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Micron Document

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<title>Pyramidenstumpf</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Pyramidenstumpf</h1>

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Ein Pyramidenstumpf ist ein Begriff aus der <a href="Geometrie" title="Geometrie">Geometrie</a>, der einen speziellen Typ von <a href="Polyeder" title="Polyeder">Polyedern</a> (Vielflächnern) beschreibt. Ein Pyramidenstumpf entsteht dadurch, dass man von einer <a href="Pyramide_(Geometrie)" title="Pyramide (Geometrie)">Pyramide</a> (Ausgangspyramide) <a href="Parallel_(Geometrie)" class="mw-redirect" title="Parallel (Geometrie)">parallel</a> zur <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a> an den <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelflächen</a> eine kleinere, <a href="%C3%84hnlichkeit_(Geometrie)" title="Ähnlichkeit (Geometrie)">ähnliche</a> Pyramide (Ergänzungspyramide) abschneidet.

Die beiden parallelen <a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Flächen</a> eines Pyramidenstumpfes sind zueinander ähnlich. Die größere dieser beiden Flächen bezeichnet man als <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a>, die kleinere als Deckfläche. Den <a href="Abstand" title="Abstand">Abstand</a> zwischen Grundfläche und Deckfläche nennt man die <a href="H%C3%B6he_(Geometrie)" title="Höhe (Geometrie)">Höhe</a> des Pyramidenstumpfes.

Das <a href="Volumen" title="Volumen">Volumen</a> eines Pyramidenstumpfes kann mit Hilfe der folgenden Formel berechnet werden:

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {h}{3}}\cdot \left(A_{\text{1}}+{\sqrt {A_{\text{1}}\cdot A_{\text{2}}}}+A_{\text{2}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aad0d0183ef966f67f1be75591021405db602b8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.743ex; height:5.343ex;" alt="{\displaystyle V={\frac {h}{3}}\cdot \left(A_{\text{1}}+{\sqrt {A_{\text{1}}\cdot A_{\text{2}}}}+A_{\text{2}}\right)}" loading="lazy"></dd></dl>

Dabei stehen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy"> für den <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> der <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> für die <a href="H%C3%B6he_(Geometrie)" title="Höhe (Geometrie)">Höhe</a> des Pyramidenstumpfes.

Für die aus <a href="Trapez_(Geometrie)" title="Trapez (Geometrie)">Trapezen</a> zusammengesetzte <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a> gibt es keine einfache Formel. Je schiefer – bei gleichbleibender Höhe – die Pyramide, bzw. der Pyramidenstumpf ist, desto größer ist die jeweils zugehörige Mantelfläche.

<div class="mw-heading mw-heading2"><h2 id="Volumen">Volumen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Herleitung_mit_Hilfe_der_zentrischen_Streckung">Herleitung mit Hilfe der zentrischen Streckung</h3></div>

Für die Berechnung des Volumens des Pyramidenstumpfes werden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14e8880a2e4243a2fe5157e574a0547ef3d5d373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}" loading="lazy"> als <a href="H%C3%B6he_(Geometrie)" title="Höhe (Geometrie)">Höhe</a> der Ausgangspyramide und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56d2d1733d09f02f33694f72b6da94681021e0f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{2}}" loading="lazy"> als Höhe der Ergänzungspyramide definiert, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}-h_{2}=h}">

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<annotation encoding="application/x-tex">{\displaystyle h_{1}-h_{2}=h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3878f6be70e1b38d8c020834a40a833948335b57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.064ex; height:2.509ex;" alt="{\displaystyle h_{1}-h_{2}=h}" loading="lazy"> gilt. Aus der <a href="Zentrische_Streckung" title="Zentrische Streckung">zentrischen Streckung</a> folgt, dass

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<annotation encoding="application/x-tex">{\displaystyle {\frac {h_{1}}{h_{2}}}=k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10916b94184c28d3c0c65ceebc8da44cc17ebfce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.539ex; height:5.843ex;" alt="{\displaystyle {\frac {h_{1}}{h_{2}}}=k}" loading="lazy"> und daher auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A_{1}}{A_{2}}}=k^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A_{1}}{A_{2}}}=k^{2}}</annotation>

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Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> der Streckfaktor der zentrischen Streckung.

Das <a href="Volumen" title="Volumen">Volumen</a> des Pyramidenstumpfes ergibt sich aus der Differenz zwischen dem Volumen der Ausgangspyramide und dem Volumen der Ergänzungspyramide:

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<annotation encoding="application/x-tex">{\displaystyle V=V_{1}-V_{2}={\frac {A_{1}\cdot h_{1}}{3}}-{\frac {A_{2}\cdot h_{2}}{3}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba8c4a46317c1d1cbbbe17729fdead10aae3f7a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.895ex; height:5.343ex;" alt="{\displaystyle V=V_{1}-V_{2}={\frac {A_{1}\cdot h_{1}}{3}}-{\frac {A_{2}\cdot h_{2}}{3}}}" loading="lazy">.</dd></dl>

Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {h_{1}}{h_{2}}}=k}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {h_{1}}{h_{2}}}=k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10916b94184c28d3c0c65ceebc8da44cc17ebfce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.539ex; height:5.843ex;" alt="{\displaystyle {\frac {h_{1}}{h_{2}}}=k}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A_{1}}{A_{2}}}=k^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A_{1}}{A_{2}}}=k^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dfa1d8650277c5f0a480b8b091b10dac089d7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.997ex; height:5.843ex;" alt="{\displaystyle {\frac {A_{1}}{A_{2}}}=k^{2}}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {h_{1}}{h_{2}}}={\frac {\sqrt {A_{1}}}{\sqrt {A_{2}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {h_{1}}{h_{2}}}={\frac {\sqrt {A_{1}}}{\sqrt {A_{2}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa5253e0dac5dd3b4dfe00081653edaa771c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.285ex; height:7.509ex;" alt="{\displaystyle {\frac {h_{1}}{h_{2}}}={\frac {\sqrt {A_{1}}}{\sqrt {A_{2}}}}}" loading="lazy">.

Die Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {h_{1}}{\sqrt {A_{1}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {h_{1}}{\sqrt {A_{1}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d95f271561a045cc3e9cbb0433ba3eceb609564.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.411ex; height:6.676ex;" alt="{\displaystyle \lambda ={\frac {h_{1}}{\sqrt {A_{1}}}}}" loading="lazy"> ergibt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}=\lambda \cdot {\sqrt {A_{1}}}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{1}=\lambda \cdot {\sqrt {A_{1}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/812f83c6a306de75d1afa69c9a1117216c00abe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.647ex; height:3.343ex;" alt="{\displaystyle h_{1}=\lambda \cdot {\sqrt {A_{1}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{2}=\lambda \cdot {\sqrt {A_{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{2}=\lambda \cdot {\sqrt {A_{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/171ae7b7183b1d2d74a68cf8ecd2a3180a1e1fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.647ex; height:3.343ex;" alt="{\displaystyle h_{2}=\lambda \cdot {\sqrt {A_{2}}}}" loading="lazy">.

Einsetzen in die Gleichung ergibt

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {A_{1}\cdot \lambda \cdot {\sqrt {A_{1}}}}{3}}-{\frac {A_{2}\cdot \lambda \cdot {\sqrt {A_{2}}}}{3}}={\frac {\lambda \cdot ({A_{1}}^{\frac {3}{2}}-{A_{2}}^{\frac {3}{2}})}{3}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39a12dea796277034ebcb82b2d6fa964e99fcb78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.347ex; height:6.843ex;" alt="{\displaystyle V={\frac {A_{1}\cdot \lambda \cdot {\sqrt {A_{1}}}}{3}}-{\frac {A_{2}\cdot \lambda \cdot {\sqrt {A_{2}}}}{3}}={\frac {\lambda \cdot ({A_{1}}^{\frac {3}{2}}-{A_{2}}^{\frac {3}{2}})}{3}}}" loading="lazy"></dd></dl>

Mit Hilfe der Formel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{3}-b^{3}=(a-b)\cdot (a^{2}+a\cdot b+b^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle a^{3}-b^{3}=(a-b)\cdot (a^{2}+a\cdot b+b^{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1fff12d1973e2628bcf098ef527166b5f5993a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.563ex; height:3.176ex;" alt="{\displaystyle a^{3}-b^{3}=(a-b)\cdot (a^{2}+a\cdot b+b^{2})}" loading="lazy"> angewendet auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {A_{1}}}}">

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<annotation encoding="application/x-tex">{\displaystyle a={\sqrt {A_{1}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbb1e06f156dac08fa96298a4af4cd44f9c1fb15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.449ex; height:3.343ex;" alt="{\displaystyle a={\sqrt {A_{1}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\sqrt {A_{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle b={\sqrt {A_{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a5b7f43c6d69e297a82a5754af49c9321a8798e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.217ex; height:3.343ex;" alt="{\displaystyle b={\sqrt {A_{2}}}}" loading="lazy"> ist das Volumen

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {\lambda }{3}}\cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)\cdot \left({\sqrt {A_{1}}}^{2}+{\sqrt {A_{1}}}\cdot {\sqrt {A_{2}}}+{\sqrt {A_{2}}}^{2}\right)={\frac {\lambda }{3}}\cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)\cdot \left(A_{1}+{\sqrt {A_{1}\cdot A_{2}}}+A_{2}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6aa3ea4d46f6c8e660dfe340445bc8cc615129d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:105.239ex; height:5.343ex;" alt="{\displaystyle V={\frac {\lambda }{3}}\cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)\cdot \left({\sqrt {A_{1}}}^{2}+{\sqrt {A_{1}}}\cdot {\sqrt {A_{2}}}+{\sqrt {A_{2}}}^{2}\right)={\frac {\lambda }{3}}\cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)\cdot \left(A_{1}+{\sqrt {A_{1}\cdot A_{2}}}+A_{2}\right)}" loading="lazy"></dd></dl>

Der Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda \cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66591e67242161cdb526a18db59998d50324e28e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.247ex; height:3.343ex;" alt="{\displaystyle \lambda \cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)}" loading="lazy"> ist die Höhe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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<msqrt>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \lambda \cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)=\lambda \cdot {\sqrt {A_{1}}}-\lambda \cdot {\sqrt {A_{2}}}=h_{1}-h_{2}=h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28ab64ba49aa0c683669c964316b5fe4b766af3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.659ex; height:3.343ex;" alt="{\displaystyle \lambda \cdot \left({\sqrt {A_{1}}}-{\sqrt {A_{2}}}\right)=\lambda \cdot {\sqrt {A_{1}}}-\lambda \cdot {\sqrt {A_{2}}}=h_{1}-h_{2}=h}" loading="lazy"></dd></dl>

Daraus ergibt sich

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {h}{3}}\cdot \left(A_{1}+{\sqrt {A_{1}\cdot A_{2}}}+A_{2}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d47e50cddec3dc7868770963a00039f69c4e2d54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.743ex; height:5.343ex;" alt="{\displaystyle V={\frac {h}{3}}\cdot \left(A_{1}+{\sqrt {A_{1}\cdot A_{2}}}+A_{2}\right)}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {A_{1}\cdot A_{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {A_{1}\cdot A_{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5592b6b75912afc797970fe05e98ba2e9ff47692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.598ex; height:3.343ex;" alt="{\displaystyle {\sqrt {A_{1}\cdot A_{2}}}}" loading="lazy"> als <a href="Geometrisches_Mittel" title="Geometrisches Mittel">geometrischem Mittel</a> der <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalte</a> von Grundfläche und Deckfläche.

<div class="mw-heading mw-heading3"><h3 id="Herleitung_mit_Hilfe_der_Integralrechnung">Herleitung mit Hilfe der Integralrechnung</h3></div>

Die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">-Achse sei so definiert, dass sie durch die Spitze der Ausgangspyramide und Ergänzungspyramide verläuft und <a href="Orthogonalit%C3%A4t" title="Orthogonalität">orthogonal</a> zur <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a> und zur Deckfläche ist. Dann ist die Höhe des Pyramidenstumpfes Teil der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">-Achse. Bezeichnet man den <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> der Schicht im <a href="Abstand" title="Abstand">Abstand</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle A(y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/def9f9d8e3ab72e445fe55858014fa8e880954b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.708ex; height:2.843ex;" alt="{\displaystyle A(y)}" loading="lazy">, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> eine <a href="Reellwertige_Funktion#Reelle_Funktion" title="Reellwertige Funktion">reelle Funktion</a> mit der Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Aus den Eigenschaften der <a href="Zentrische_Streckung" title="Zentrische Streckung">zentrischen Streckung</a> kann man eine Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(y)}">

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<annotation encoding="application/x-tex">{\displaystyle A(y)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A(y)}{A_{1}}}={\frac {y^{2}}{h_{1}^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84b9dd4178048ca5cd26ed844833358dc31c1efa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.872ex; height:6.676ex;" alt="{\displaystyle {\frac {A(y)}{A_{1}}}={\frac {y^{2}}{h_{1}^{2}}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A(y)}{A_{2}}}={\frac {y^{2}}{h_{2}^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1eca017deb0f4c7afa0d0582385b24c8110837fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.872ex; height:6.676ex;" alt="{\displaystyle {\frac {A(y)}{A_{2}}}={\frac {y^{2}}{h_{2}^{2}}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle A(y)={\frac {A_{1}}{h_{1}^{2}}}\cdot y^{2}={\frac {A_{2}}{h_{2}^{2}}}\cdot y^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20c4628f3af8d96c06de92638ac0c9786e2db3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.959ex; height:6.343ex;" alt="{\displaystyle A(y)={\frac {A_{1}}{h_{1}^{2}}}\cdot y^{2}={\frac {A_{2}}{h_{2}^{2}}}\cdot y^{2}}" loading="lazy"></dd></dl>

Es gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {h_{2}^{2}}{h_{1}^{2}}}={\frac {A_{2}}{A_{1}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {h_{2}^{2}}{h_{1}^{2}}}={\frac {A_{2}}{A_{1}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a881c631ea7456fe9b1ee9262d22fbf1702255a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.961ex; height:6.843ex;" alt="{\displaystyle {\frac {h_{2}^{2}}{h_{1}^{2}}}={\frac {A_{2}}{A_{1}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {h_{2}}{h_{1}}}={\frac {\sqrt {A_{2}}}{\sqrt {A_{1}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {h_{2}}{h_{1}}}={\frac {\sqrt {A_{2}}}{\sqrt {A_{1}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee058f2c7391c1c0f7a62421ea88c8af17ea620e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.285ex; height:7.509ex;" alt="{\displaystyle {\frac {h_{2}}{h_{1}}}={\frac {\sqrt {A_{2}}}{\sqrt {A_{1}}}}}" loading="lazy">.

Daraus ergibt sich das <a href="Volumen" title="Volumen">Volumen</a> des Pyramidenstumpfes als <a href="Integralrechnung" title="Integralrechnung">Integral</a> der <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(y)}">

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<annotation encoding="application/x-tex">{\displaystyle A(y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/def9f9d8e3ab72e445fe55858014fa8e880954b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.708ex; height:2.843ex;" alt="{\displaystyle A(y)}" loading="lazy"> auf dem <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [h_{2},h_{1}]}">

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<annotation encoding="application/x-tex">{\displaystyle [h_{2},h_{1}]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba35e0ee8198219fe2bfce6607b805ceda45e4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.114ex; height:2.843ex;" alt="{\displaystyle [h_{2},h_{1}]}" loading="lazy"> nach dem <a href="Prinzip_von_Cavalieri" title="Prinzip von Cavalieri">Prinzip von Cavalieri</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V&=\int _{h_{2}}^{h_{1}}A(y)\ \mathrm {d} y=\int _{h_{2}}^{h_{1}}{\frac {A_{1}}{h_{1}^{2}}}\cdot y^{2}\ \mathrm {d} y={\frac {A_{1}}{h_{1}^{2}}}\cdot \int _{h_{2}}^{h_{1}}y^{2}\ \mathrm {d} y\\&={\frac {A_{1}}{h_{1}^{2}}}\cdot \left[{\frac {1}{3}}\cdot y^{3}\right]_{h_{2}}^{h_{1}}={\frac {A_{1}}{h_{1}^{2}}}\cdot \left({\frac {1}{3}}\cdot h_{1}^{3}-{\frac {1}{3}}\cdot h_{2}^{3}\right)={\frac {1}{3}}\cdot {\frac {A_{1}}{h_{1}^{2}}}\cdot (h_{1}^{3}-h_{2}^{3})={\frac {1}{3}}\cdot {\frac {A_{1}}{h_{1}^{2}}}\cdot (h_{1}-h_{2})\cdot (h_{1}^{2}+h_{1}\cdot h_{2}+h_{2}^{2})\\&={\frac {h}{3}}\cdot {\frac {A_{1}}{h_{1}^{2}}}\cdot (h_{1}^{2}+h_{1}\cdot h_{2}+h_{2}^{2})={\frac {h}{3}}\cdot \left(A_{1}+A_{1}\cdot {\frac {h_{2}}{h_{1}}}+A_{1}\cdot {\frac {h_{2}^{2}}{h_{1}^{2}}}\right)={\frac {h}{3}}\cdot \left(A_{1}+{\sqrt {A_{1}\cdot A_{2}}}+A_{2}\right)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/968663e6812be6b766d08a26d6cf11f6db6e6e39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:111.593ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}V&=\int _{h_{2}}^{h_{1}}A(y)\ \mathrm {d} y=\int _{h_{2}}^{h_{1}}{\frac {A_{1}}{h_{1}^{2}}}\cdot y^{2}\ \mathrm {d} y={\frac {A_{1}}{h_{1}^{2}}}\cdot \int _{h_{2}}^{h_{1}}y^{2}\ \mathrm {d} y\\&={\frac {A_{1}}{h_{1}^{2}}}\cdot \left[{\frac {1}{3}}\cdot y^{3}\right]_{h_{2}}^{h_{1}}={\frac {A_{1}}{h_{1}^{2}}}\cdot \left({\frac {1}{3}}\cdot h_{1}^{3}-{\frac {1}{3}}\cdot h_{2}^{3}\right)={\frac {1}{3}}\cdot {\frac {A_{1}}{h_{1}^{2}}}\cdot (h_{1}^{3}-h_{2}^{3})={\frac {1}{3}}\cdot {\frac {A_{1}}{h_{1}^{2}}}\cdot (h_{1}-h_{2})\cdot (h_{1}^{2}+h_{1}\cdot h_{2}+h_{2}^{2})\\&={\frac {h}{3}}\cdot {\frac {A_{1}}{h_{1}^{2}}}\cdot (h_{1}^{2}+h_{1}\cdot h_{2}+h_{2}^{2})={\frac {h}{3}}\cdot \left(A_{1}+A_{1}\cdot {\frac {h_{2}}{h_{1}}}+A_{1}\cdot {\frac {h_{2}^{2}}{h_{1}^{2}}}\right)={\frac {h}{3}}\cdot \left(A_{1}+{\sqrt {A_{1}\cdot A_{2}}}+A_{2}\right)\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Grenzfälle">Grenzfälle</h3></div>

Nähern sich <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grund-</a> und Deckfläche einem <a href="Kreis" title="Kreis">Kreis</a>, erhält man einen <a href="Kegelstumpf" title="Kegelstumpf">Kegelstumpf</a>, für den dieselbe allgemeine Volumenformel gilt. Geht die <a href="H%C3%B6he_(Geometrie)" title="Höhe (Geometrie)">Höhe</a> der Ausgangspyramide dagegen gegen <a href="Unendlichkeit" title="Unendlichkeit">unendlich</a>, nähert sich der <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> der Deckfläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"> dem der <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy"> und man erhält ein <a href="Prisma_(Geometrie)" title="Prisma (Geometrie)">Prisma</a>, dessen Volumenformel sich damit wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}=A_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}=A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ccb4687bbe819cd8e1f6c6afb06b9316a6c18b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.693ex; height:2.509ex;" alt="{\displaystyle A_{1}=A_{2}}" loading="lazy"> zu der Formel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=h\cdot A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=h\cdot A_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66b2be28639ffbb02314d0a10c706e9c21fd21b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.701ex; height:2.509ex;" alt="{\displaystyle V=h\cdot A_{1}}" loading="lazy"> vereinfacht. Geht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"> schließlich gegen 0, erhält man ja nachdem, ob die Grundfläche ein n-Eck oder Kreis ist, eine komplette <a href="Pyramide_(Geometrie)" title="Pyramide (Geometrie)">Pyramide</a> oder einen <a href="Kegel_(Geometrie)" title="Kegel (Geometrie)">Kegel</a> mit der allgemeinen Volumenformel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\textstyle {\frac {h}{3}}\cdot A_{1}}">

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\textstyle {\frac {h}{3}}\cdot A_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b9e70351335db0ef37b567f210d53689e1fa035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.145ex; height:3.843ex;" alt="{\displaystyle V=\textstyle {\frac {h}{3}}\cdot A_{1}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Regelmäßiger_Pyramidenstumpf">Regelmäßiger Pyramidenstumpf</h2></div>

Ein regelmäßiger Pyramidenstumpf hat jeweils ein <a href="Regelm%C3%A4%C3%9Figes_Vieleck" class="mw-redirect" title="Regelmäßiges Vieleck">regelmäßiges Vieleck</a> als <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a> und als Deckfläche. Die <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a> besteht aus <a href="Kongruenz_(Geometrie)" title="Kongruenz (Geometrie)">kongruenten</a> <a href="Gleichschenkliges_Trapez" class="mw-redirect" title="Gleichschenkliges Trapez">gleichschenkligen Trapezen</a>. Der <a href="Mittelpunkt" title="Mittelpunkt">Mittelpunkt</a> der Deckfläche liegt senkrecht über dem Mittelpunkt der Grundfläche.

<div class="mw-heading mw-heading3"><h3 id="Formeln">Formeln</h3></div>

<th colspan="3" style="background:#C0C0FF">Größen eines regelmäßigen Pyramidenstumpfs (regelmäßiges n-Eck mit Seitenlänge a1 als Grundfläche, regelmäßiges n-Eck mit Seitenlänge a2 als Deckfläche und Höhe h)

</th></tr>

<tr>

</td>

<td class="hintergrundfarbe5">Allgemeiner Fall

</td>

<td class="hintergrundfarbe5">Quadratischer Pyramidenstumpf

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Volumen" title="Volumen">Volumen</a>

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V={\frac {n\cdot (a_{1}^{3}-a_{2}^{3})\cdot h}{12\cdot (a_{1}-a_{2})}}\cdot \cot \left({\frac {\pi }{n}}\right)={\frac {n\cdot (a_{1}^{2}+a_{1}\cdot a_{2}+a_{2}^{2})\cdot h}{12}}\cdot \cot \left({\frac {\pi }{n}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21ecbdbe122186377c5e1ff06b954d920eb59a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:68.933ex; height:6.843ex;" alt="{\displaystyle V={\frac {n\cdot (a_{1}^{3}-a_{2}^{3})\cdot h}{12\cdot (a_{1}-a_{2})}}\cdot \cot \left({\frac {\pi }{n}}\right)={\frac {n\cdot (a_{1}^{2}+a_{1}\cdot a_{2}+a_{2}^{2})\cdot h}{12}}\cdot \cot \left({\frac {\pi }{n}}\right)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V={\frac {(a_{1}^{3}-a_{2}^{3})\cdot h}{3\cdot (a_{1}-a_{2})}}={\frac {(a_{1}^{2}+a_{1}\cdot a_{2}+a_{2}^{2})\cdot h}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9c5bd60480524cc9a964a7a9cf486a7ec51eb8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.216ex; height:6.843ex;" alt="{\displaystyle V={\frac {(a_{1}^{3}-a_{2}^{3})\cdot h}{3\cdot (a_{1}-a_{2})}}={\frac {(a_{1}^{2}+a_{1}\cdot a_{2}+a_{2}^{2})\cdot h}{3}}}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Oberflächeninhalt</a>

</td>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A={\frac {n}{4}}\cdot \left((a_{1}^{2}+a_{2}^{2})\cdot \cot \left({\frac {\pi }{n}}\right)+(a_{1}+a_{2})\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}\cdot \cot ^{2}\left({\frac {\pi }{n}}\right)}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a770619c3ed5f454fd0c607e5fbe7672356a888f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:76.423ex; height:6.176ex;" alt="{\displaystyle A={\frac {n}{4}}\cdot \left((a_{1}^{2}+a_{2}^{2})\cdot \cot \left({\frac {\pi }{n}}\right)+(a_{1}+a_{2})\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}\cdot \cot ^{2}\left({\frac {\pi }{n}}\right)}}\right)}" loading="lazy">

</td>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=a_{1}^{2}+a_{2}^{2}+(a_{1}+a_{2})\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/324d21cf67bec247d24d98d9d3fc9b064443e8cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:46.658ex; height:4.843ex;" alt="{\displaystyle A=a_{1}^{2}+a_{2}^{2}+(a_{1}+a_{2})\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}}}}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> der <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a>

</td>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}={\frac {n\cdot a_{1}^{2}}{4}}\cdot \cot \left({\frac {\pi }{n}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/893a8f1d7c3b45bb6fb12b6bce122754c6944d4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.875ex; height:5.843ex;" alt="{\displaystyle A_{1}={\frac {n\cdot a_{1}^{2}}{4}}\cdot \cot \left({\frac {\pi }{n}}\right)}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}=a_{1}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/501135529c28b48d51c62e785e98fd038bec4856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.18ex; height:3.176ex;" alt="{\displaystyle A_{1}=a_{1}^{2}}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5">Flächeninhalt der Deckfläche

</td>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}={\frac {n\cdot a_{2}^{2}}{4}}\cdot \cot \left({\frac {\pi }{n}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dc9d69fd3270e409130c11d0b39ff9fdfd555a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.875ex; height:5.843ex;" alt="{\displaystyle A_{2}={\frac {n\cdot a_{2}^{2}}{4}}\cdot \cot \left({\frac {\pi }{n}}\right)}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}=a_{2}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4633b753dee8a9d4a8764843ea8ae49fa73c3fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.18ex; height:3.176ex;" alt="{\displaystyle A_{2}=a_{2}^{2}}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5">Flächeninhalt der <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a>

</td>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M={\frac {n\cdot (a_{1}+a_{2})}{4}}\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}\cdot \cot ^{2}\left({\frac {\pi }{n}}\right)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f635377de3d8a8c533d4ec8af2668f1a0abe6589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.858ex; height:6.343ex;" alt="{\displaystyle M={\frac {n\cdot (a_{1}+a_{2})}{4}}\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}\cdot \cot ^{2}\left({\frac {\pi }{n}}\right)}}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=(a_{1}+a_{2})\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/754a1804210fddc2a02b799dc72d36f1e3c2fa9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:37.109ex; height:4.843ex;" alt="{\displaystyle M=(a_{1}+a_{2})\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}}}}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5">Steilkantenlänge

</td>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l=\left(h^{2}+{\frac {(a_{1}-a_{2})^{2}}{4\cdot \sin ^{2}\left({\frac {\pi }{n}}\right)}}\right)^{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa08c09bf2b5556b1188d08cf2737489a5be772a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.982ex; height:8.676ex;" alt="{\displaystyle l=\left(h^{2}+{\frac {(a_{1}-a_{2})^{2}}{4\cdot \sin ^{2}\left({\frac {\pi }{n}}\right)}}\right)^{\frac {1}{2}}}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l={\sqrt {h^{2}+{\frac {(a_{1}-a_{2})^{2}}{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/041cb8f5fa325718643df333971e1168c8b0b492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.457ex; height:7.676ex;" alt="{\displaystyle l={\sqrt {h^{2}+{\frac {(a_{1}-a_{2})^{2}}{2}}}}}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Umkugel" title="Umkugel">Umkugelradius</a>

</td>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{u}={\frac {1}{2\cdot h}}\cdot \left(h^{4}+{\frac {h^{2}\cdot (a_{1}^{2}+a_{2}^{2})}{2\cdot \sin ^{2}\left({\frac {\pi }{n}}\right)}}+{\frac {\left(a_{1}^{2}-a_{2}^{2}\right)^{2}}{16\cdot \sin ^{4}\left({\frac {\pi }{n}}\right)}}\right)^{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c6761aa6905cdd0960b0e3d68f3a141af002ec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:52.72ex; height:9.343ex;" alt="{\displaystyle r_{u}={\frac {1}{2\cdot h}}\cdot \left(h^{4}+{\frac {h^{2}\cdot (a_{1}^{2}+a_{2}^{2})}{2\cdot \sin ^{2}\left({\frac {\pi }{n}}\right)}}+{\frac {\left(a_{1}^{2}-a_{2}^{2}\right)^{2}}{16\cdot \sin ^{4}\left({\frac {\pi }{n}}\right)}}\right)^{\frac {1}{2}}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{u}={\frac {1}{2\cdot h}}\cdot \left(h^{4}+h^{2}\cdot (a_{1}^{2}+a_{2}^{2})+{\frac {\left(a_{1}^{2}-a_{2}^{2}\right)^{2}}{4}}\right)^{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d4424579c54126c80ed1b1f978d0224b49e2ddc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:50.611ex; height:9.343ex;" alt="{\displaystyle r_{u}={\frac {1}{2\cdot h}}\cdot \left(h^{4}+h^{2}\cdot (a_{1}^{2}+a_{2}^{2})+{\frac {\left(a_{1}^{2}-a_{2}^{2}\right)^{2}}{4}}\right)^{\frac {1}{2}}}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Innenwinkel" title="Innenwinkel">Innenwinkel</a> der <a href="Regelm%C3%A4%C3%9Figes_Polygon" title="Regelmäßiges Polygon">regelmäßigen</a> Grundfläche

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {n-2}{n}}\cdot 180^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d83ede5feee9ca56432f5b0381dc717a41fbeb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.04ex; height:5.176ex;" alt="{\displaystyle \alpha ={\frac {n-2}{n}}\cdot 180^{\circ }}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =90^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30f0fb8d6c2247b522d8bcb4a62b5c0a9f9ee7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.965ex; height:2.343ex;" alt="{\displaystyle \alpha =90^{\circ }}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5">Basiswinkel der <a href="Gleichschenkliges_Trapez" class="mw-redirect" title="Gleichschenkliges Trapez">gleichschenkligen Trapeze</a>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{1}=\alpha _{2}=\arctan \left({\frac {1}{a_{1}-a_{2}}}\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}\cdot \cot ^{2}\left({\frac {\pi }{n}}\right)}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/055d2b2a69aa6b74675826cfc16c522c5fe86208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.602ex; height:6.176ex;" alt="{\displaystyle \alpha _{1}=\alpha _{2}=\arctan \left({\frac {1}{a_{1}-a_{2}}}\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}\cdot \cot ^{2}\left({\frac {\pi }{n}}\right)}}\right)}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{1}=\alpha _{2}=\arctan \left({\frac {1}{a_{1}-a_{2}}}\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e21a584cbb06e9b2d6d917005e081fe6f99e4cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.763ex; height:6.176ex;" alt="{\displaystyle \alpha _{1}=\alpha _{2}=\arctan \left({\frac {1}{a_{1}-a_{2}}}\cdot {\sqrt {4\cdot h^{2}+(a_{1}-a_{2})^{2}}}\right)}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Winkel" title="Winkel">Winkel</a> zwischen Grundfläche und gleichschenkligen Trapezen

</td>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{1}=\arctan \left({\frac {2\cdot h\cdot \tan \left({\frac {\pi }{n}}\right)}{a_{1}-a_{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47d0628a36a81e675f8959219ed1d705eafac65a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.623ex; height:7.509ex;" alt="{\displaystyle \beta _{1}=\arctan \left({\frac {2\cdot h\cdot \tan \left({\frac {\pi }{n}}\right)}{a_{1}-a_{2}}}\right)}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{1}=\arctan \left({\frac {2\cdot h}{a_{1}-a_{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3185fae629a5b5652a2f4a8e9fbc63064c47e984.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.6ex; height:6.176ex;" alt="{\displaystyle \beta _{1}=\arctan \left({\frac {2\cdot h}{a_{1}-a_{2}}}\right)}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Diederwinkel" title="Diederwinkel">Diederwinkel</a> zwischen den gleichschenkligen Trapezen

</td>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{2}=2\cdot \arctan \left({\frac {1}{2\cdot h}}\cdot \left({\frac {4\cdot h^{2}\cdot \sin ^{2}\left({\frac {\pi }{n}}\right)+(a_{1}-a_{2})^{2}}{\tan ^{2}\left({\frac {\pi }{n}}\right)-\sin ^{2}\left({\frac {\pi }{n}}\right)}}\right)^{\frac {1}{2}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98069a67560c6739c0e9a86c69975cd8154cfedb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:59.68ex; height:9.843ex;" alt="{\displaystyle \beta _{2}=2\cdot \arctan \left({\frac {1}{2\cdot h}}\cdot \left({\frac {4\cdot h^{2}\cdot \sin ^{2}\left({\frac {\pi }{n}}\right)+(a_{1}-a_{2})^{2}}{\tan ^{2}\left({\frac {\pi }{n}}\right)-\sin ^{2}\left({\frac {\pi }{n}}\right)}}\right)^{\frac {1}{2}}\right)}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{2}=2\cdot \arctan \left({\frac {1}{2\cdot h}}\cdot {\sqrt {4\cdot h^{2}+2\cdot (a_{1}-a_{2})^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c1325a039cab8cf83699723a4081130cff8b14f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.406ex; height:6.176ex;" alt="{\displaystyle \beta _{2}=2\cdot \arctan \left({\frac {1}{2\cdot h}}\cdot {\sqrt {4\cdot h^{2}+2\cdot (a_{1}-a_{2})^{2}}}\right)}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Winkel" title="Winkel">Winkel</a> zwischen Kante und Grundfläche

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =\arctan \left({\frac {2\cdot h\cdot \sin \left({\frac {\pi }{n}}\right)}{a_{1}-a_{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/684ed2eaddbfb9e839bcf4c2127628a77794926d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.012ex; height:7.509ex;" alt="{\displaystyle \gamma =\arctan \left({\frac {2\cdot h\cdot \sin \left({\frac {\pi }{n}}\right)}{a_{1}-a_{2}}}\right)}" loading="lazy">

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =\arctan \left({\frac {2\cdot h}{{\sqrt {2}}\cdot \left(a_{1}-a_{2}\right)}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b637e7a45e67a468ef65731c9be32b7be70cb6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.34ex; height:7.509ex;" alt="{\displaystyle \gamma =\arctan \left({\frac {2\cdot h}{{\sqrt {2}}\cdot \left(a_{1}-a_{2}\right)}}\right)}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Raumwinkel" title="Raumwinkel">Raumwinkel</a> in den Ecken der Grundfläche

</td>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<msqrt>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega _{1}=4\cdot \arctan \left({\sqrt {\tan \left({\frac {2\cdot \alpha _{1}+\alpha }{4}}\right)\cdot \tan \left({\frac {2\cdot \alpha _{1}-\alpha }{4}}\right)\cdot \tan ^{2}\left({\frac {\alpha }{4}}\right)}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0182b1bdca06bb295efa3726d49e9b268cc13963.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.672ex; height:7.509ex;" alt="{\displaystyle \Omega _{1}=4\cdot \arctan \left({\sqrt {\tan \left({\frac {2\cdot \alpha _{1}+\alpha }{4}}\right)\cdot \tan \left({\frac {2\cdot \alpha _{1}-\alpha }{4}}\right)\cdot \tan ^{2}\left({\frac {\alpha }{4}}\right)}}\right)}" loading="lazy">

</td></tr>

<tr>

<td class="hintergrundfarbe5"><a href="Raumwinkel" title="Raumwinkel">Raumwinkel</a> in den Ecken der Deckfläche

</td>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<msqrt>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega _{2}=4\cdot \arctan \left({\sqrt {\tan \left({\frac {2\cdot (\pi -\alpha _{1})+\alpha }{4}}\right)\cdot \tan \left({\frac {2\cdot (\pi -\alpha _{1})-\alpha }{4}}\right)\cdot \tan ^{2}\left({\frac {\alpha }{4}}\right)}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64bf4efdb679ff311b26a28eabf7dcd4aab620fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:80.635ex; height:7.676ex;" alt="{\displaystyle \Omega _{2}=4\cdot \arctan \left({\sqrt {\tan \left({\frac {2\cdot (\pi -\alpha _{1})+\alpha }{4}}\right)\cdot \tan \left({\frac {2\cdot (\pi -\alpha _{1})-\alpha }{4}}\right)\cdot \tan ^{2}\left({\frac {\alpha }{4}}\right)}}\right)}" loading="lazy">

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Kegelstumpf" title="Kegelstumpf">Kegelstumpf</a></li>

<li><a href="Doppelpyramide" title="Doppelpyramide">Doppelpyramide</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Rolf Baumann: <cite style="font-style:italic">Geometrie für die 9./10. Klasse</cite>. Zentrische Streckung, Satz des Pythagoras, Kreis- und Körperberechnungen. 4. Auflage. Mentor-Verlag, München 2003, ISBN 3-580-63635-9, S. 95 ff.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Pyramidenstumpf&rft.au=Rolf+Baumann&rft.btitle=Geometrie+f%C3%BCr+die+9.%2F10.+Klasse&rft.date=2003&rft.edition=4.&rft.genre=book&rft.isbn=3580636359&rft.pages=95+ff&rft.place=M%C3%BCnchen&rft.pub=Mentor-Verlag" style="display:none"> </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Frustums_of_pyramids?uselang=de">Commons: Pyramidenstumpf</a> – Sammlung von Bildern, Videos und Audiodateien</div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><a href="https://de.wiktionary.org/wiki/Pyramidenstumpf" class="extiw external" title="wikt:Pyramidenstumpf">Wiktionary: Pyramidenstumpf</a> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div>

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PyramidalFrustum.html">Pyramidal Frustum.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li>

<li><a rel="nofollow" class="external text" href="https://rechneronline.de/pi/pyramidenstumpf.php">Pyramidenstumpf-Rechner</a> (Web Application zum Berechnen)</li>

<li><a rel="nofollow" class="external text" href="https://www.handwerk-technik.de/_files_media/probeseiten/5615_01.pdf">Pyramidenstumpf</a> (Beispielaufgaben)</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-10-03" href="https://de.wikipedia.org/wiki/?title=Pyramidenstumpf&oldid=260281305">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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