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<title>Parallelepiped</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Parallelepiped</span></h1>

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<p>Ein <b>Parallelepiped</b> oder <b>Spat</b> (früher auch <i>Parallelflach</i>) ist ein <a href="Geometrie" title="Geometrie">geometrischer</a> <a href="K%C3%B6rper_(Geometrie)" title="Körper (Geometrie)">Körper</a>, der von sechs <a href="Parallelogramm" title="Parallelogramm">Parallelogrammen</a> begrenzt wird, von denen je zwei gegenüberliegende <a href="Kongruenz_(Geometrie)" title="Kongruenz (Geometrie)">kongruent</a> (deckungsgleich) sind und in <a href="Parallel_(Geometrie)" class="mw-redirect" title="Parallel (Geometrie)">parallelen</a> <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebenen</a> liegen.

</p><p>Ein Parallelepiped hat 12 Kanten, von denen je 4 parallel verlaufen und untereinander gleich lang sind, und 8 Ecken, in denen diese Kanten in maximal 3 verschiedenen Winkeln zueinander zusammenlaufen.

</p><p><a href="Quader" title="Quader">Quader</a>, bei denen alle Winkel gleich 90° sind, und <a href="Rhomboeder" title="Rhomboeder">Rhomboeder</a>, bei denen alle Kanten gleich lang und 3 <a href="Innenwinkel" title="Innenwinkel">Innenwinkel</a> gleich sind, sind Spezialfälle des Parallelepipeds. Der <a href="W%C3%BCrfel_(Geometrie)" title="Würfel (Geometrie)">Würfel</a> vereinigt beide Spezialfälle in einer Figur. Das Parallelepiped ist ein spezielles <a href="Prisma_(Geometrie)" title="Prisma (Geometrie)">Prisma</a> mit einem Parallelogramm als <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a>.

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

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

<p>Stellt man die drei an einer <a href="Ecke" title="Ecke">Ecke</a> zusammentreffende Kanten als <a href="Vektor" title="Vektor">Vektoren</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/befa941df6751fd0d377ccd7d31130ef372a3b8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.615ex; height:3.176ex;" alt="{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}" loading="lazy"></span> dar, so ergibt sich das <a href="Volumen" title="Volumen">Volumen</a> des Parallelepipeds aus dem Betrag des <a href="Spatprodukt" title="Spatprodukt">Spatproduktes</a> (gemischtes <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> und <a href="Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a>). Das Volumen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> ist das Produkt der <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> (<a href="Parallelogramm" title="Parallelogramm">Parallelogramm</a>) und der <a href="H%C3%B6he_(Geometrie)" title="Höhe (Geometrie)">Höhe</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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</math></span><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"></span> des Parallelepipeds. Mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=|{\vec {a}}|\cdot |{\vec {b}}|\cdot \sin(\gamma )=|{\vec {a}}\times {\vec {b}}|}">

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<annotation encoding="application/x-tex">{\displaystyle G=|{\vec {a}}|\cdot |{\vec {b}}|\cdot \sin(\gamma )=|{\vec {a}}\times {\vec {b}}|}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7437ea517d7cebb5412e0de116ebbb8dd55593e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.678ex; height:3.343ex;" alt="{\displaystyle G=|{\vec {a}}|\cdot |{\vec {b}}|\cdot \sin(\gamma )=|{\vec {a}}\times {\vec {b}}|}" loading="lazy"></span>, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span> der <a href="Winkel" title="Winkel">Winkel</a> zwischen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/546e6615827e17295718741fd0b86f639a947f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c9ef58be7103eb0b2bfcb460df23430f6a36216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" loading="lazy"></span> ist, und der Höhe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=|{\vec {c}}|\cdot |\cos(\theta )|}">

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<annotation encoding="application/x-tex">{\displaystyle h=|{\vec {c}}|\cdot |\cos(\theta )|}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e33e36227af77029889eae75ac4cc73df8b50e06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.325ex; height:2.843ex;" alt="{\displaystyle h=|{\vec {c}}|\cdot |\cos(\theta )|}" loading="lazy"></span>, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> der Winkel zwischen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/965bd8710781b710cbfdb79da0b4e3b097bef506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}" loading="lazy"></span> und dem <a href="Normalenvektor" title="Normalenvektor">Normalenvektor</a> auf der Grundfläche ist, ergibt sich

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V&=G\cdot h=(|{\vec {a}}|\cdot |{\vec {b}}|\cdot \sin(\gamma ))\cdot |{\vec {c}}|\cdot |\cos(\theta )|=|{\vec {a}}\times {\vec {b}}|\cdot |{\vec {c}}|\cdot |\cos(\theta )|\\&=|({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}|\end{aligned}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb405a036287b6d0a15af5cbd076805924b0f1c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.276ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}V&=G\cdot h=(|{\vec {a}}|\cdot |{\vec {b}}|\cdot \sin(\gamma ))\cdot |{\vec {c}}|\cdot |\cos(\theta )|=|{\vec {a}}\times {\vec {b}}|\cdot |{\vec {c}}|\cdot |\cos(\theta )|\\&=|({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}|\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Das gemischte Produkt nennt man <a href="Spatprodukt" title="Spatprodukt">Spatprodukt</a>. Es kann als <a href="Determinante" title="Determinante">Determinante</a> geschrieben werden. Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}=(a_{1},a_{2},a_{3})^{T},\quad {\vec {b}}=(b_{1},b_{2},b_{3})^{T},\quad {\vec {c}}=(c_{1},c_{2},c_{3})^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(a_{1},a_{2},a_{3})^{T},\quad {\vec {b}}=(b_{1},b_{2},b_{3})^{T},\quad {\vec {c}}=(c_{1},c_{2},c_{3})^{T}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e41914bc6436b0485040ebc5daa0bda795a4a193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.545ex; height:3.343ex;" alt="{\displaystyle {\vec {a}}=(a_{1},a_{2},a_{3})^{T},\quad {\vec {b}}=(b_{1},b_{2},b_{3})^{T},\quad {\vec {c}}=(c_{1},c_{2},c_{3})^{T}}" loading="lazy"></span> ist das <a href="Volumen" title="Volumen">Volumen</a> dann:

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<annotation encoding="application/x-tex">{\displaystyle V=\left|\det {\begin{pmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{pmatrix}}\;\right|}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/883c8dda96169a71e856f7b265b19e21c66480b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.302ex; height:9.176ex;" alt="{\displaystyle V=\left|\det {\begin{pmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{pmatrix}}\;\right|}" loading="lazy"></span></dd></dl>

<p>Eine nur von den <a href="Geometrisch" class="mw-redirect" title="Geometrisch">geometrischen</a> Eigenschaften (Kantenlängen, Winkel zwischen benachbarten Kanten) abhängige <a href="Formel" title="Formel">Formel</a> für das <a href="Volumen" title="Volumen">Volumen</a> ist:

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<annotation encoding="application/x-tex">{\displaystyle V=a\cdot b\cdot c\cdot {\sqrt {1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/323c5e18b3f0fdc0553be53da51efd3d9a693cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:75.054ex; height:4.843ex;" alt="{\displaystyle V=a\cdot b\cdot c\cdot {\sqrt {1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}}" loading="lazy"></span></dd></dl>

<p>Dabei sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d410eee0a783a3d1af61e7d7b0649f4a0ca437c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.748ex; height:3.343ex;" alt="{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}" loading="lazy"></span> die <a href="Winkel" title="Winkel">Winkel</a> zwischen den Kanten und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">

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</p><p>Der Nachweis dieser Formel lässt sich mit den <a href="Determinante#Eigenschaften" title="Determinante">Eigenschaften einer Determinante</a> und der geometrischen Deutung des <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukts</a> führen. Es sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V^{2}&=(\det(M))^{2}=\det(M)\cdot \det(M)=\det(M^{T})\cdot \det(M)=\det(M^{T}\cdot M)\\&=\det {\begin{pmatrix}{\vec {a}}\cdot {\vec {a}}&{\vec {a}}\cdot {\vec {b}}&{\vec {a}}\cdot {\vec {c}}\\{\vec {b}}\cdot {\vec {a}}&{\vec {b}}\cdot {\vec {b}}&{\vec {b}}\cdot {\vec {c}}\\{\vec {c}}\cdot {\vec {a}}&{\vec {c}}\cdot {\vec {b}}&{\vec {c}}\cdot {\vec {c}}\end{pmatrix}}=a^{2}\cdot b^{2}\cdot c^{2}\cdot (1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma ))\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98e155897a3f650bd2580d06d7de3f14281aae05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:108.09ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}V^{2}&=(\det(M))^{2}=\det(M)\cdot \det(M)=\det(M^{T})\cdot \det(M)=\det(M^{T}\cdot M)\\&=\det {\begin{pmatrix}{\vec {a}}\cdot {\vec {a}}&{\vec {a}}\cdot {\vec {b}}&{\vec {a}}\cdot {\vec {c}}\\{\vec {b}}\cdot {\vec {a}}&{\vec {b}}\cdot {\vec {b}}&{\vec {b}}\cdot {\vec {c}}\\{\vec {c}}\cdot {\vec {a}}&{\vec {c}}\cdot {\vec {b}}&{\vec {c}}\cdot {\vec {c}}\end{pmatrix}}=a^{2}\cdot b^{2}\cdot c^{2}\cdot (1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma ))\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Im letzten Schritt wurden die <a href="Gleichung" title="Gleichung">Gleichungen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}\cdot {\vec {a}}=a^{2},\quad {\vec {b}}\cdot {\vec {b}}=b^{2},\quad {\vec {c}}\cdot {\vec {c}}=c^{2},\quad {\vec {a}}\cdot {\vec {b}}=a\cdot b\cdot \cos(\gamma ),\quad {\vec {a}}\cdot {\vec {c}}=a\cdot c\cdot \cos(\beta ),\quad {\vec {b}}\cdot {\vec {c}}=b\cdot c\cdot \cos(\alpha )}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\cdot {\vec {a}}=a^{2},\quad {\vec {b}}\cdot {\vec {b}}=b^{2},\quad {\vec {c}}\cdot {\vec {c}}=c^{2},\quad {\vec {a}}\cdot {\vec {b}}=a\cdot b\cdot \cos(\gamma ),\quad {\vec {a}}\cdot {\vec {c}}=a\cdot c\cdot \cos(\beta ),\quad {\vec {b}}\cdot {\vec {c}}=b\cdot c\cdot \cos(\alpha )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24126184dc5541734820bd5146d03da781ac4b5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.418ex; height:3.343ex;" alt="{\displaystyle {\vec {a}}\cdot {\vec {a}}=a^{2},\quad {\vec {b}}\cdot {\vec {b}}=b^{2},\quad {\vec {c}}\cdot {\vec {c}}=c^{2},\quad {\vec {a}}\cdot {\vec {b}}=a\cdot b\cdot \cos(\gamma ),\quad {\vec {a}}\cdot {\vec {c}}=a\cdot c\cdot \cos(\beta ),\quad {\vec {b}}\cdot {\vec {c}}=b\cdot c\cdot \cos(\alpha )}" loading="lazy"></span> benutzt.

</p>

<div class="mw-heading mw-heading3"><h3 id="Oberfläche"><span id="Oberfl.C3.A4che"></span>Oberfläche</h3></div>

<p>Der <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> der Oberfläche ergibt sich aus der Summe der Flächeninhalte der einzelnen <a href="Seitenfl%C3%A4che" class="mw-redirect" title="Seitenfläche">Seitenflächen</a>, den 6 <a href="Parallelogramm" title="Parallelogramm">Parallelogrammen</a>:

</p>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=2\cdot \left(|{\vec {a}}\times {\vec {b}}|+|{\vec {a}}\times {\vec {c}}|+|{\vec {b}}\times {\vec {c}}|\right)\\&=2\cdot a\cdot b\cdot \sin(\gamma )+2\cdot a\cdot c\cdot \sin(\beta )+2\cdot b\cdot c\cdot \sin(\alpha )\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36270f5a4e86d1e1578e974920e23b9a07da75d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:54.418ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}A&=2\cdot \left(|{\vec {a}}\times {\vec {b}}|+|{\vec {a}}\times {\vec {c}}|+|{\vec {b}}\times {\vec {c}}|\right)\\&=2\cdot a\cdot b\cdot \sin(\gamma )+2\cdot a\cdot c\cdot \sin(\beta )+2\cdot b\cdot c\cdot \sin(\alpha )\end{aligned}}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Flächenwinkel"><span id="Fl.C3.A4chenwinkel"></span>Flächenwinkel</h3></div>

<p>In der <a href="Ecke" title="Ecke">Ecke</a>, in der die <a href="Vektor" title="Vektor">Vektoren</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/befa941df6751fd0d377ccd7d31130ef372a3b8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.615ex; height:3.176ex;" alt="{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}" loading="lazy"></span> zusammentreffen, liegen die <a href="Innenwinkel" title="Innenwinkel">Innenwinkel</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d410eee0a783a3d1af61e7d7b0649f4a0ca437c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.748ex; height:3.343ex;" alt="{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}" loading="lazy"></span>. Diese Ecke bildet zusammen mit den 3 benachbarten Ecken ein <a href="Tetraeder" title="Tetraeder">Tetraeder</a>. Betrachtet man die <a href="Umkugel" title="Umkugel">Umkugel</a> dieses Tetraeders, dann gilt nach dem <a href="Kosinussatz#Kosinussatz_für_Kugeldreiecke" title="Kosinussatz">Kosinussatz für Kugeldreiecke</a> die <a href="Gleichung" title="Gleichung">Gleichung</a>

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(\alpha )=\cos(\beta )\cdot \cos(\gamma )+\sin(\beta )\cdot \sin(\gamma )\cdot \cos(\beta _{a})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adfd0027cf3ae7ddbc7a6dcce1a45faa23361a4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.082ex; height:2.843ex;" alt="{\displaystyle \cos(\alpha )=\cos(\beta )\cdot \cos(\gamma )+\sin(\beta )\cdot \sin(\gamma )\cdot \cos(\beta _{a})}" loading="lazy"></span></dd></dl>

<p>Dabei ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{a}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d48576ab8a7b331bf471250cb91846ec78a7243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.509ex;" alt="{\displaystyle \beta _{a}}" loading="lazy"></span> der <a href="Fl%C3%A4chenwinkel" class="mw-redirect" title="Flächenwinkel">Flächenwinkel</a> zwischen den beiden <a href="Seitenfl%C3%A4che" class="mw-redirect" title="Seitenfläche">Seitenflächen</a>, die am <a href="Vektor" title="Vektor">Vektor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</p><p>Daraus folgt

</p>

<msub>

</mrow>

</msub>

<mi>arccos</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{a}=\arccos \left({\frac {\cos(\alpha )-\cos(\beta )\cdot \cos(\gamma )}{\sin(\beta )\cdot \sin(\gamma )}}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a4de4fa1ccd9696a2754b65f4dc50a059113605.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.354ex; height:6.509ex;" alt="{\displaystyle \beta _{a}=\arccos \left({\frac {\cos(\alpha )-\cos(\beta )\cdot \cos(\gamma )}{\sin(\beta )\cdot \sin(\gamma )}}\right)}" loading="lazy"></span></dd></dl>

<p>Die Flächenwinkel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{b}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ebb7b4cea705dfdfe4b4db9e02303189bae9d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.253ex; height:2.509ex;" alt="{\displaystyle \beta _{b}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{c}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f956d5fae99fba9f88da2ac6d9e0d515e613ecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.26ex; height:2.509ex;" alt="{\displaystyle \beta _{c}}" loading="lazy"></span> ergeben sich entsprechend.

</p>

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

<p>Der <a href="Raumwinkel" title="Raumwinkel">Raumwinkel</a> in der <a href="Ecke" title="Ecke">Ecke</a> eines <a href="Polyeder" title="Polyeder">Polyeders</a> kann mit dem Satz von L’Huilier berechnet werden.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Für den Raumwinkel, der in der Ecke mit den <a href="Innenwinkel" title="Innenwinkel">Innenwinkeln</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =\angle ({\vec {b}},{\vec {c}}),\quad \beta =\angle ({\vec {a}},{\vec {c}}),\quad \gamma =\angle ({\vec {a}},{\vec {b}})}</annotation>

</semantics>

</p>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mi>arctan</mi>

<mrow>

<msqrt>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mi>arctan</mi>

<mrow>

<msqrt>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Omega _{1}&=4\cdot \arctan \left({\sqrt {\tan \left({\frac {\theta _{s}}{2}}\right)\cdot \tan \left({\frac {\theta _{s}-\theta _{a}}{2}}\right)\cdot \tan \left({\frac {\theta _{s}-\theta _{b}}{2}}\right)\cdot \tan \left({\frac {\theta _{s}-\theta _{c}}{2}}\right)}}\right)\\&=4\cdot \arctan \left({\sqrt {\tan \left({\frac {\alpha +\beta +\gamma }{4}}\right)\cdot \tan \left({\frac {-\alpha +\beta +\gamma }{4}}\right)\cdot \tan \left({\frac {\alpha -\beta +\gamma }{4}}\right)\cdot \tan \left({\frac {\alpha +\beta -\gamma }{4}}\right)}}\right)\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc6e1f8c0896fcada43c579a0551df0c57936bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:98.259ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\Omega _{1}&=4\cdot \arctan \left({\sqrt {\tan \left({\frac {\theta _{s}}{2}}\right)\cdot \tan \left({\frac {\theta _{s}-\theta _{a}}{2}}\right)\cdot \tan \left({\frac {\theta _{s}-\theta _{b}}{2}}\right)\cdot \tan \left({\frac {\theta _{s}-\theta _{c}}{2}}\right)}}\right)\\&=4\cdot \arctan \left({\sqrt {\tan \left({\frac {\alpha +\beta +\gamma }{4}}\right)\cdot \tan \left({\frac {-\alpha +\beta +\gamma }{4}}\right)\cdot \tan \left({\frac {\alpha -\beta +\gamma }{4}}\right)\cdot \tan \left({\frac {\alpha +\beta -\gamma }{4}}\right)}}\right)\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{s}={\frac {\alpha +\beta +\gamma }{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{s}={\frac {\alpha +\beta +\gamma }{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b063a20245bcd83eada8588ed651d6dc82ba6b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.791ex; height:5.509ex;" alt="{\displaystyle \theta _{s}={\frac {\alpha +\beta +\gamma }{2}}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{a}=\alpha }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{a}=\alpha }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1f50a8846d80b83d08f4e9a80340e6d686cf165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.778ex; height:2.509ex;" alt="{\displaystyle \theta _{a}=\alpha }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{b}=\beta }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{b}=\beta }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d23f96509df0d4ad6ccca50558a2e7edeb53fc2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.459ex; height:2.509ex;" alt="{\displaystyle \theta _{b}=\beta }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{c}=\gamma }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{c}=\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edc0be4a50618c4faa4a18d26d6149b64e0bb72d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.395ex; height:2.676ex;" alt="{\displaystyle \theta _{c}=\gamma }" loading="lazy"></span> ist.

</p><p>Zwei diagonal gegenüber liegende <a href="Raumwinkel" title="Raumwinkel">Raumwinkel</a> in Ecken des Parallelepipeds sind jeweils gleich, weil die 3 anliegenden Innenwinkel gleich sind. Die anderen drei Raumwinkel ergeben sich für

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{a}=\alpha ,\quad \theta _{b}=180^{\circ }-\beta ,\quad \theta _{c}=180^{\circ }-\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6797c3ae816c53300079b12cab2a1f90ec9b6f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.109ex; height:2.843ex;" alt="{\displaystyle \theta _{a}=\alpha ,\quad \theta _{b}=180^{\circ }-\beta ,\quad \theta _{c}=180^{\circ }-\gamma }" loading="lazy"></span></li>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{a}=180^{\circ }-\alpha ,\quad \theta _{b}=\beta ,\quad \theta _{c}=180^{\circ }-\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0aaaf84b4c9dda4446e050523b014bf897273398.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.109ex; height:2.843ex;" alt="{\displaystyle \theta _{a}=180^{\circ }-\alpha ,\quad \theta _{b}=\beta ,\quad \theta _{c}=180^{\circ }-\gamma }" loading="lazy"></span></li>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{a}=180^{\circ }-\alpha ,\quad \theta _{b}=180^{\circ }-\beta ,\quad \theta _{c}=\gamma }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac5a5d8a8d77e96848c9157183acd08d1b810134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.109ex; height:2.843ex;" alt="{\displaystyle \theta _{a}=180^{\circ }-\alpha ,\quad \theta _{b}=180^{\circ }-\beta ,\quad \theta _{c}=\gamma }" loading="lazy"></span></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Tabelle:_Zusammenfassung">Tabelle: Zusammenfassung</h3></div>

<th colspan="3" style="background:#C0C0FF">Größen eines Parallelepipeds mit den Kantenlängen <i>a</i>, <i>b</i>, <i>c</i> und den Innenwinkeln <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

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

</semantics>

</th></tr>

<tr>

<td class="hintergrundfarbe5"><b>Parallelepiped</b>

</td>

</li>

</li>

</ul>

</td></tr>

<tr>

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

</td>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

</semantics>

</td></tr>

<tr>

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

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=2\cdot a\cdot b\cdot \sin(\gamma )+2\cdot a\cdot c\cdot \sin(\beta )+2\cdot b\cdot c\cdot \sin(\alpha )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a477ccc9aa489d57cbe7fa724a4dac91b9782d2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.667ex; height:2.843ex;" alt="{\displaystyle A=2\cdot a\cdot b\cdot \sin(\gamma )+2\cdot a\cdot c\cdot \sin(\beta )+2\cdot b\cdot c\cdot \sin(\alpha )}" loading="lazy"></span>

</td></tr>

<tr>

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h={\frac {a}{\sin(\alpha )}}\cdot {\sqrt {1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2ca6fa4fb1ab7f53d79d2ff8dd026a270558122.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:75.002ex; height:5.843ex;" alt="{\displaystyle h={\frac {a}{\sin(\alpha )}}\cdot {\sqrt {1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}}" loading="lazy"></span>

</td></tr>

<tr>

<td class="hintergrundfarbe5"><b><a href="Diagonale_(Geometrie)" title="Diagonale (Geometrie)">Raumdiagonale</a></b><br>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |{\vec {a}}+{\vec {b}}+{\vec {c}}|}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0d9fadddcac211c47324ab93ba1a21e9574143c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.521ex; height:3.343ex;" alt="{\displaystyle |{\vec {a}}+{\vec {b}}+{\vec {c}}|}" loading="lazy"></span>

</p>

</td>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d={\sqrt {a^{2}+b^{2}+c^{2}+2\cdot a\cdot b\cdot \sin(\gamma )+2\cdot a\cdot c\cdot \sin(\beta )+2\cdot b\cdot c\cdot \sin(\alpha )}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97a1de67b7088a4b8a5f5709ee729853f537dc2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:70.381ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {a^{2}+b^{2}+c^{2}+2\cdot a\cdot b\cdot \sin(\gamma )+2\cdot a\cdot c\cdot \sin(\beta )+2\cdot b\cdot c\cdot \sin(\alpha )}}}" loading="lazy"></span>

</td></tr>

<tr>

<td class="hintergrundfarbe5"><b>Winkel zwischen</b>

<p><b>benachbarten Flächen</b>

</p>

</td>

<msub>

</mrow>

</msub>

<mi>arccos</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{a}=\arccos \left({\frac {\cos(\alpha )-\cos(\beta )\cdot \cos(\gamma )}{\sin(\beta )\cdot \sin(\gamma )}}\right)}</annotation>

</semantics>

</td></tr>

<tr>

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

</td>

<msub>

</mrow>

</msub>

<mi>arctan</mi>

<mrow>

<msqrt>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82dab6e2e46c61fc03b728d80d716e12c5ff4e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:76.858ex; height:6.176ex;" alt="{\displaystyle \Omega _{1}=4\cdot \arctan \left({\sqrt {\tan \left({\tfrac {\alpha +\beta +\gamma }{4}}\right)\cdot \tan \left({\tfrac {-\alpha +\beta +\gamma }{4}}\right)\cdot \tan \left({\tfrac {\alpha -\beta +\gamma }{4}}\right)\cdot \tan \left({\tfrac {\alpha +\beta -\gamma }{4}}\right)}}\right)}" loading="lazy"></span>

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

<div class="mw-heading mw-heading2"><h2 id="Raumfüllung_mit_Parallelepipeden"><span id="Raumf.C3.BCllung_mit_Parallelepipeden"></span>Raumfüllung mit Parallelepipeden</h2></div>

<p>Der <a href="Dreidimensional" class="mw-redirect" title="Dreidimensional">dreidimensionale</a> <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidische Raum</a> kann lückenlos mit <a href="Kongruenz_(Geometrie)" title="Kongruenz (Geometrie)">kongruenten</a> Parallelepipeden ausgefüllt werden. Solche dreidimensionalen <a href="Parkettierung" title="Parkettierung">Parkettierungen</a> werden <i><a href="Raumf%C3%BCllung" title="Raumfüllung">Raumfüllung</a></i> genannt.

</p><p>Diese Raumfüllung aus Parallelepipeden bildet ein <a href="Gitter_(Geometrie)" title="Gitter (Geometrie)">Gitter</a>. Dieses Gitter enthält <a href="Parallel_(Geometrie)" class="mw-redirect" title="Parallel (Geometrie)">parallele</a> <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebenen</a>. Die im Gitter benachbarten <a href="Raumwinkel" title="Raumwinkel">Raumwinkel</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b95a2e6dbc4b6f2d03b3397691609aa9544230f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Omega _{1}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/789466adc27ae2abb1feb4e8a00d374d8c6521da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Omega _{2}}" loading="lazy"></span> entsprechen zusammen dem <a href="Fl%C3%A4chenwinkel" class="mw-redirect" title="Flächenwinkel">Flächenwinkel</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{a}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d48576ab8a7b331bf471250cb91846ec78a7243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.509ex;" alt="{\displaystyle \beta _{a}}" loading="lazy"></span>. Der volle Flächenwinkel beträgt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot \pi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e292d13c8b12fd1fa50db8ac2ac404366b7c3c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.174ex; height:2.176ex;" alt="{\displaystyle 2\cdot \pi }" loading="lazy"></span> und der volle Raumwinkel beträgt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot \pi \ \mathrm {sr} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\cdot \pi \ \mathrm {sr} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b09682bf0922393ae490de8fe5e0d5719bb25a2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.582ex; height:2.176ex;" alt="{\displaystyle 4\cdot \pi \ \mathrm {sr} }" loading="lazy"></span>. Daher gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{a}={\frac {\Omega _{1}+\Omega _{2}}{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{a}={\frac {\Omega _{1}+\Omega _{2}}{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c92b939d66678f5cdb7ee8867fb9ba1c5d14aa83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.657ex; height:5.343ex;" alt="{\displaystyle \beta _{a}={\frac {\Omega _{1}+\Omega _{2}}{2}}}" loading="lazy"></span>. Entsprechend gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{b}={\frac {\Omega _{1}+\Omega _{3}}{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{b}={\frac {\Omega _{1}+\Omega _{3}}{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a02bf755aad75089ccc52fea10da549f95f5890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.493ex; height:5.343ex;" alt="{\displaystyle \beta _{b}={\frac {\Omega _{1}+\Omega _{3}}{2}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{c}={\frac {\Omega _{1}+\Omega _{4}}{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{c}={\frac {\Omega _{1}+\Omega _{4}}{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccae44aecc5fc222d85ef9564fb6d70c4a4066e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.5ex; height:5.343ex;" alt="{\displaystyle \beta _{c}={\frac {\Omega _{1}+\Omega _{4}}{2}}}" loading="lazy"></span>.

</p><p>In den Gitterpunkten treffen 8 Raumwinkel zusammen und bilden einen vollen Raumwinkel, wobei 2 diagonal gegenüber liegende Raumwinkel jeweils gleich sind. Es gilt also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot \Omega _{1}+2\cdot \Omega _{2}+2\cdot \Omega _{3}+2\cdot \Omega _{4}=4\cdot \pi \ \mathrm {sr} }">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle 2\cdot \Omega _{1}+2\cdot \Omega _{2}+2\cdot \Omega _{3}+2\cdot \Omega _{4}=4\cdot \pi \ \mathrm {sr} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f9230b04217a8f59eddcab2d8223d0503666fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.497ex; height:2.509ex;" alt="{\displaystyle 2\cdot \Omega _{1}+2\cdot \Omega _{2}+2\cdot \Omega _{3}+2\cdot \Omega _{4}=4\cdot \pi \ \mathrm {sr} }" loading="lazy"></span>.

</p>

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

<p>Das <a href="Parallelotop" title="Parallelotop">Parallelotop</a> beziehungsweise <i>n</i>-Parallelotop ist eine Verallgemeinerung des Parallelepipeds im <i>n</i>-<a href="Dimensionalit%C3%A4t" class="mw-redirect" title="Dimensionalität">dimensionalen</a> <a href="Raum_(Mathematik)" title="Raum (Mathematik)">Raum</a>. Allerdings wird auch diese Verallgemeinerung selbst auch als (n-dimensionales) Parallelepiped bezeichnet und die Begriffe Parallelotop, Parallelflach und Parallelepiped werden oft synonym verwendet. Das <a href="Zweidimensional" class="mw-redirect" title="Zweidimensional">zweidimensionale</a> Parallelotop ist das <a href="Parallelogramm" title="Parallelogramm">Parallelogramm</a>.

</p><p>Ein <i>n</i>-Parallelotop ist das <a href="Bild_(Mathematik)" title="Bild (Mathematik)">Bild</a> des <a href="W%C3%BCrfel_(Geometrie)" title="Würfel (Geometrie)">Einheitswürfels</a> unter einer <a href="Affine_Abbildung" title="Affine Abbildung">affinen Abbildung</a>. Der Einheitswürfel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle I^{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f154f60f0493b12f8bdb8e8815b3153ed6ffd938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.343ex;" alt="{\displaystyle I^{n}}" loading="lazy"></span> ist eine <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> von <a href="Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkten</a>, deren <a href="Koordinatensystem" title="Koordinatensystem">Koordinaten</a> einen Wert zwischen 0 und 1 annehmen, das heißt

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle I^{n}:=\left\{(x_{1},\dots ,x_{n})\mid 0\leq x_{i}\leq 1\right\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3436db8792b412eb2388e75a5ac6a7a3ac968e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.013ex; height:2.843ex;" alt="{\displaystyle I^{n}:=\left\{(x_{1},\dots ,x_{n})\mid 0\leq x_{i}\leq 1\right\}}" loading="lazy"></span></dd></dl>

<p>Das Parallelotop ist ein konvexes <a href="Polytop_(Geometrie)" title="Polytop (Geometrie)">Polytop</a> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}">

<msup>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8226f30650ee4fe4e640c6d2798127e80e9c160d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}" loading="lazy"></span> <a href="Ecke" title="Ecke">Ecken</a>. Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m<n}">

</mstyle>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/490c01b0cb770144f28afd17bb5fef277daf6f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle m<n}" loading="lazy"></span> sind seine <i>m</i>-dimensionalen Seiten selbst <i>m</i>-dimensionale Parallelotope.

</p>

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

<ul><li>Konrad Königsberger: <i>Analysis</i>. Band 2. Springer, Berlin 2004, ISBN 3-540-20389-3.</li></ul>

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

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

<li><a href="Rhomboeder" title="Rhomboeder">Rhomboeder</a></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"><span class="mw-default-size" typeof="mw:File"><span title="Commons"></span></span></div><b><span class=""><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Parallelepipeds?uselang=de"><span lang="en">Commons</span>: Parallelepipeds</a></span></b> – Sammlung von Bildern, Videos und Audiodateien</div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"></span></span></span><b><a href="https://de.wiktionary.org/wiki/Parallelepiped" class="extiw external" title="wikt:Parallelepiped">Wiktionary: Parallelepiped</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div>

<ul><li><a rel="nofollow" class="external text" href="https://de.serlo.org/mathe/geometrie/analytische-geometrie/flaechen-volumenberechnung/volumenberechnung-der-analytischen-geometrie/parallelotop">Definition des Parallelepipeds</a> aus den Mathematik-Vorlesungen der Universität Stuttgart, abgerufen am 6. Dezember 2020</li>

<li><a rel="nofollow" class="external text" href="https://learnattack.de/schuelerlexikon/mathematik/parallelepiped">Formeln zum Parallelepiped</a> aus dem Duden-Schülerlexikon, abgerufen am 6. Dezember 2020</li>

<li><a rel="nofollow" class="external text" href="https://www.mein-lernen.at/mathematik2/vektoren-raum/vektoren-volumen/parallelepiped">Formeln, Beispiele und Übungen zum Parallelepiped</a> aus <i>mein-lernen.at</i>, abgerufen am 6. Dezember 2020</li>

<li><a rel="nofollow" class="external text" href="https://rechneronline.de/pi/parallelepiped.php">Parallelepiped-Rechner</a> aus <i>rechneronline.de</i>, abgerufen am 6. Dezember 2020</li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Wolfram MathWorld: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SphericalExcess.html">Spherical Excess</a></span>

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-07-06" href="https://de.wikipedia.org/wiki/?title=Parallelepiped&oldid=257655463">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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