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Micron Document

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<title>Kugelsegment</title>

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

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<p>Ein <b>Kugelsegment</b> oder <b>Kugelabschnitt</b> ist ein Teil einer <a href="Kugel" title="Kugel">Kugel</a>, der durch den Schnitt mit einer <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> abgetrennt wird. Die kreisförmige Schnittfläche ist seine <a href="Grundfl%C3%A4che_(Geometrie)" title="Grundfläche (Geometrie)">Grundfläche</a>. Ein Kugelsegment ist ein Sonderfall einer <a href="Kugelschicht" title="Kugelschicht">Kugelschicht</a>, bei der die Höhe bis an die Kugeloberfläche heranreicht. Eine <a href="Halbkugel" class="mw-redirect" title="Halbkugel">Halbkugel</a> ist wiederum ein Sonderfall eines Kugelsegments, bei der die Schnittebene den Kugelmittelpunkt enthält. Der gekrümmte Teil der Oberfläche eines Kugelsegments wird <b>Kugelkalotte</b>, auch <b>Kugelkappe</b> oder <b>Kugelhaube</b> genannt.<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>

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

<p>Für die Berechnung von <a href="Volumen" title="Volumen">Volumen</a>, <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a> und <a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Oberfläche</a> eines Kugelsegments gelten die folgenden <a href="Mathematische_Formel" class="mw-redirect" title="Mathematische Formel">Formeln</a>. Dabei bezeichnet <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 r}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> den <a href="Radius" title="Radius">Radius</a> der <a href="Kugel" title="Kugel">Kugel</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 a}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> den Radius des Basiskreises des Kugelsegments 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 h}">

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

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</p><p>Diese drei Größen sind nicht unabhängig voneinander. Das Kugelsegment ist durch zwei beliebige dieser drei Größen bestimmt. Aus zwei der drei Größen lässt sich die dritte nach dem Satz des Pythagoras berechnen

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a21d7fd8fd01179d4a318a84a47f71b8ef68befa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.417ex; height:3.176ex;" alt="{\displaystyle (r-h)^{2}+a^{2}=r^{2}}" loading="lazy"></span>, bzw. <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 2rh=a^{2}+h^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 2rh=a^{2}+h^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/546f3ebe275e56a98f1b82167a17e0a060779d26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.166ex; height:2.843ex;" alt="{\displaystyle 2rh=a^{2}+h^{2}}" loading="lazy"></span></dd></dl>

<p>In den folgenden Formeln ist bei ± Minus zu nehmen, wenn das Kugelsegment weniger als die halbe Kugel groß ist, sonst Plus.

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<annotation encoding="application/x-tex">{\displaystyle h=r\pm {\sqrt {r^{2}-a^{2}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150a66afd05d3bef10af15c60ce769e1a741ac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.878ex; height:3.509ex;" alt="{\displaystyle h=r\pm {\sqrt {r^{2}-a^{2}}}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle h^{2}=2r\,(r\pm {\sqrt {r^{2}-a^{2}}})-a^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67ac8722fcf80b91803488a7e64be7b29a3b65a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.464ex; height:3.509ex;" alt="{\displaystyle h^{2}=2r\,(r\pm {\sqrt {r^{2}-a^{2}}})-a^{2}}" loading="lazy"></span></dd></dl>

<p>Statt <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}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle 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 h}">

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

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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> reicht auch die Angabe des <a href="Winkel" title="Winkel">Winkels</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 \theta _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{0}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18b67de6bf25dba7a24e66967ff6319858798734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{0}}" loading="lazy"></span> des Basiskreises (siehe Abbildung). Es gilt:

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<annotation encoding="application/x-tex">{\displaystyle a=r\,\sin \theta _{0}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ba2a0184c81d490a062268c22b3d188f00832c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.539ex; height:2.509ex;" alt="{\displaystyle a=r\,\sin \theta _{0}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle h=r\,(1-\cos \theta _{0})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a0c0f28e70deb687244a652622b7a6f6d089678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.328ex; height:2.843ex;" alt="{\displaystyle h=r\,(1-\cos \theta _{0})}" loading="lazy"></span></dd></dl>

<p>Es gibt deshalb jeweils mehrere Formeln, je nachdem, welche der Größen gegeben sind.

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<caption>Größen eines Kugelsegments mit dem Radius <i>r</i> der Kugel,<br>dem Radius <i>a</i> des Basiskreises und der Höhe <i>h</i>

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<td rowspan="4"><b><a href="Volumen" title="Volumen">Volumen</a></b>

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<annotation encoding="application/x-tex">{\displaystyle V_{r,h}={\frac {\pi }{3}}h^{2}\,(3r-h)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/beeacf2b6f7c42e9a37f894eb0bab99fe52520ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.98ex; height:4.676ex;" alt="{\displaystyle V_{r,h}={\frac {\pi }{3}}h^{2}\,(3r-h)}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle V_{h,a}={\frac {\pi }{6}}h\,(3a^{2}+h^{2})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7749fd3325172139ed6cb09970769eb50673ae4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.343ex; height:4.676ex;" alt="{\displaystyle V_{h,a}={\frac {\pi }{6}}h\,(3a^{2}+h^{2})}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle V_{r,a}={\frac {\pi }{3}}\,(r\pm {\sqrt {r^{2}-a^{2}}})\,\left(a^{2}+r\,(r\pm {\sqrt {r^{2}-a^{2}}})\right)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9168ec9d8d9e27be2c72af1211e16da806eb1520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.918ex; height:4.843ex;" alt="{\displaystyle V_{r,a}={\frac {\pi }{3}}\,(r\pm {\sqrt {r^{2}-a^{2}}})\,\left(a^{2}+r\,(r\pm {\sqrt {r^{2}-a^{2}}})\right)}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle V_{r,\theta _{0}}={\frac {\pi }{3}}\,r^{3}\,(2+\cos \theta _{0})\,(1-\cos \theta _{0})^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c1b6222249c79bcba4a0e34564fabcb482718f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.884ex; height:4.676ex;" alt="{\displaystyle V_{r,\theta _{0}}={\frac {\pi }{3}}\,r^{3}\,(2+\cos \theta _{0})\,(1-\cos \theta _{0})^{2}}" loading="lazy"></span>

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<td rowspan="5"><b><a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt der</a></b><br><b><a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Oberfläche</a></b><br> Anteil der Basisfläche <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 =\pi a^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle O_{r,h,a}=\pi \,(2rh+a^{2})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle O_{r,h}=\pi \,h(4r-h)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle O_{r,a}=\pi \left(a^{2}+2r\,(r\pm {\sqrt {r^{2}-a^{2}}})\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle O_{r,\theta _{0}}=2\pi \,r^{2}\,(1-\cos \theta _{0}+{\tfrac {1}{2}}\sin ^{2}\theta _{0})}</annotation>

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<td rowspan="4"><b><a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt der</a> </b><br><b><a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a></b>

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<annotation encoding="application/x-tex">{\displaystyle M_{r,h}=2\pi \,rh}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle M_{h,a}=\pi \,(a^{2}+h^{2})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle M_{r,a}=2\pi \,r\,\left(r\pm {\sqrt {r^{2}-a^{2}}}\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle M_{r,\theta _{0}}=2\pi \,r^{2}\,(1-\cos \theta _{0})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6dec1677faf2d18008aa9cfe066fea363c8118e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.213ex; height:3.343ex;" alt="{\displaystyle M_{r,\theta _{0}}=2\pi \,r^{2}\,(1-\cos \theta _{0})}" loading="lazy"></span>

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

<div class="mw-heading mw-heading3"><h3 id="Sonderfälle"><span id="Sonderf.C3.A4lle"></span>Sonderfälle</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d85effe873c5417b69c424b6d9167db79db3976c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.486ex; height:2.176ex;" alt="{\displaystyle h=r}" loading="lazy"></span> 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 a=r}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e91859146bb04493cc06c66b5caa648eb5dfc2b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.377ex; height:1.676ex;" alt="{\displaystyle a=r}" loading="lazy"></span> und das Kugelsegment eine <a href="Halbkugel" class="mw-redirect" title="Halbkugel">Halbkugel</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 V={\tfrac {2\pi }{3}}r^{3},\ M=2\pi r^{2},\ O=3\pi r^{2}.}">

<mfrac>

<mrow>

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</mfrac>

</mstyle>

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<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle V={\tfrac {2\pi }{3}}r^{3},\ M=2\pi r^{2},\ O=3\pi r^{2}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e456f254f9c467721b196d86db0a4b1a2a779509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:33.072ex; height:3.676ex;" alt="{\displaystyle V={\tfrac {2\pi }{3}}r^{3},\ M=2\pi r^{2},\ O=3\pi r^{2}.}" loading="lazy"></span>

</p><p>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 h=2r}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7676287a8e0acccb4ab5614ee8b7f522e0408a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.649ex; height:2.176ex;" alt="{\displaystyle h=2r}" loading="lazy"></span> 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 a=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90d476e5e765a5d77bbcff32e4584579207ec7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" loading="lazy"></span> und das Kugelsegment ist eine ganze <a href="Kugel" title="Kugel">Kugel</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 V={\tfrac {4\pi }{3}}r^{3},\ M=O=4\pi r^{2}.}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V={\tfrac {4\pi }{3}}r^{3},\ M=O=4\pi r^{2}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4e37987ce6a65a69d4438f6771c7dbb1caaa5f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.86ex; height:3.676ex;" alt="{\displaystyle V={\tfrac {4\pi }{3}}r^{3},\ M=O=4\pi r^{2}.}" loading="lazy"></span>

</p>

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

<p>Nach dem <a href="Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a> 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 (r-h)^{2}+a^{2}=r^{2}}">

<msup>

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</msup>

<msup>

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</msup>

<msup>

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</msup>

</mstyle>

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

</semantics>

</p>

<msup>

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</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2rh=a^{2}+h^{2}}</annotation>

</semantics>

<p>Das <a href="Volumen" title="Volumen">Volumen</a> eines Kugelsegments ergibt sich aus dem Volumenintegral für <a href="Rotationsk%C3%B6rper" title="Rotationskörper">Rotationskörper</a> für den <a href="Kreisbogen" title="Kreisbogen">Kreisbogen</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 y=f(x)={\sqrt {r^{2}-(x-r)^{2}}}={\sqrt {2rx-x^{2}}}}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

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</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=f(x)={\sqrt {r^{2}-(x-r)^{2}}}={\sqrt {2rx-x^{2}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c95ed3b4563cd995570a2ee1183494bbd30faa56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:41.307ex; height:4.843ex;" alt="{\displaystyle y=f(x)={\sqrt {r^{2}-(x-r)^{2}}}={\sqrt {2rx-x^{2}}}}" loading="lazy"></span>:

</p>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\pi \int \limits _{0}^{h}f(x)^{2}\,dx=\pi \int \limits _{0}^{h}2rx-x^{2}\,dx={\frac {\pi \,h^{2}}{3}}\,(3r-h)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/686ba688d8002fbfb6624443a2ca0d8bb2b1dd18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:53.32ex; height:9.343ex;" alt="{\displaystyle V=\pi \int \limits _{0}^{h}f(x)^{2}\,dx=\pi \int \limits _{0}^{h}2rx-x^{2}\,dx={\frac {\pi \,h^{2}}{3}}\,(3r-h)}" loading="lazy"></span>.</dd></dl>

<p>Entsprechend ergibt sich die <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a> eines Kugelsegments (ohne Basiskreis) aus der Flächenformel für <a href="Rotationsfl%C3%A4che" title="Rotationsfläche">Rotationsflächen</a>

</p>

</mrow>

</mrow>

</munderover>

<msqrt>

<msup>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=2\pi \int \limits _{0}^{h}f(x){\sqrt {1+f'(x)^{2}}}\,dx=2\pi \,r\int \limits _{0}^{h}dx=2\pi \,rh}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ce88fe9fd4f85cbe963794362a70f88bdca84c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:51.788ex; height:9.343ex;" alt="{\displaystyle M=2\pi \int \limits _{0}^{h}f(x){\sqrt {1+f'(x)^{2}}}\,dx=2\pi \,r\int \limits _{0}^{h}dx=2\pi \,rh}" loading="lazy"></span> .</dd></dl>

<p>Und mit Basiskreis: <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 O=\pi \,(2rh+a^{2})=\pi \,(2a^{2}+h^{2})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O=\pi \,(2rh+a^{2})=\pi \,(2a^{2}+h^{2})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/808db668aa43c72930b9d6e5c2941d0e2e448a2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.382ex; height:3.176ex;" alt="{\displaystyle O=\pi \,(2rh+a^{2})=\pi \,(2a^{2}+h^{2})}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Höherdimensionale_euklidische_Räume"><span id="H.C3.B6herdimensionale_euklidische_R.C3.A4ume"></span>Höherdimensionale euklidische Räume</h2></div>

<p>Eine Kalotte im n-dimensionalen Raum hat Volumen und Mantelfläche<sup id="cite_ref-S-Li_2-0" class="reference"><a href="#cite_note-S-Li-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

<msup>

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</msup>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

<mfrac>

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<mrow>

<mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mi>arccos</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

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<mfrac>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{cap^{n}}={\frac {\pi ^{\frac {n-1}{2}}\,r^{n}}{\,\Gamma \left({\frac {n+1}{2}}\right)}}\int \limits _{0}^{\arccos \left({\frac {r-h}{r}}\right)}\sin ^{n}(\theta )\,\mathrm {d} \theta ={\frac {1}{2}}V_{n}\,r^{n}I_{(2rh-h^{2})/r^{2}}\left({\frac {n+1}{2}},{\frac {1}{2}}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f2f7f2e0d372d5ce538d3b933e2bc917b9d5afc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:72.425ex; height:11.343ex;" alt="{\displaystyle V_{cap^{n}}={\frac {\pi ^{\frac {n-1}{2}}\,r^{n}}{\,\Gamma \left({\frac {n+1}{2}}\right)}}\int \limits _{0}^{\arccos \left({\frac {r-h}{r}}\right)}\sin ^{n}(\theta )\,\mathrm {d} \theta ={\frac {1}{2}}V_{n}\,r^{n}I_{(2rh-h^{2})/r^{2}}\left({\frac {n+1}{2}},{\frac {1}{2}}\right)}" loading="lazy"></span></dd>

<msub>

<msup>

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</msup>

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</msub>

<mfrac>

</mfrac>

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<msub>

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<msup>

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</msup>

<msub>

<msup>

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<msup>

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</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

<mfrac>

</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{cap^{n}}={\frac {1}{2}}M_{n}\,r^{n-1}I_{(2rh-h^{2})/r^{2}}\left({\frac {n-1}{2}},{\frac {1}{2}}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6418f99219bee64b01182a99bba7ec4948e6043a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.755ex; height:6.176ex;" alt="{\displaystyle M_{cap^{n}}={\frac {1}{2}}M_{n}\,r^{n-1}I_{(2rh-h^{2})/r^{2}}\left({\frac {n-1}{2}},{\frac {1}{2}}\right)}" loading="lazy"></span></dd></dl>

<p>mit

</p>

<ul><li>der <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a> Γ</li>

<li>dem Vollvolumen <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_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ebc5a637019ce3415183f06995aeeca93547767.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle V_{n}}" loading="lazy"></span></li>

<li>dem vollen Mantel <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}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8656f32ad5c50e679b491b361a423727491496a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.473ex; height:2.509ex;" alt="{\displaystyle M_{n}}" loading="lazy"></span></li>

<li>der regularisierten <a href="Betafunktion" class="mw-disambig" title="Betafunktion">Betafunktion</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 I_{sin^{2}(\theta _{0})}(n/2,1/2)={\tfrac {B(sin^{2}(\theta _{0}),n/2,1/2)}{B(n/2,1/2)}}}">

<msub>

<msup>

</mrow>

</msup>

<msub>

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</msub>

</mrow>

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<mfrac>

<mrow>

<msup>

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<mrow>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{sin^{2}(\theta _{0})}(n/2,1/2)={\tfrac {B(sin^{2}(\theta _{0}),n/2,1/2)}{B(n/2,1/2)}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a30f673c38944a8ae17e623208b37e32b04b47a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.854ex; height:5.009ex;" alt="{\displaystyle I_{sin^{2}(\theta _{0})}(n/2,1/2)={\tfrac {B(sin^{2}(\theta _{0}),n/2,1/2)}{B(n/2,1/2)}}}" loading="lazy"></span> mit den Werten <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\phi /\pi }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d834197dbba5199a8d6fb524e4da90b0fafe2cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.042ex; height:2.843ex;" alt="{\displaystyle 2\phi /\pi }" loading="lazy"></span> für n=1, <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 1-\cos(\phi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-\cos(\phi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a2ed5871ff51e442f1099a56effeff37d7b9dc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.309ex; height:2.843ex;" alt="{\displaystyle 1-\cos(\phi )}" loading="lazy"></span> für n=2, <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\phi -\sin(2\phi )/\pi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\phi -\sin(2\phi )/\pi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ed79bd8733fb8ada78e006acdb29e007cf3d945.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.096ex; height:2.843ex;" alt="{\displaystyle 2\phi -\sin(2\phi )/\pi }" loading="lazy"></span> für n=3 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 1-3\cos(\phi )/2+\cos ^{3}(\phi )/2}">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-3\cos(\phi )/2+\cos ^{3}(\phi )/2}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3a783b0d36dab6ad0f19a5e14450ed99adaab84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.709ex; height:3.176ex;" alt="{\displaystyle 1-3\cos(\phi )/2+\cos ^{3}(\phi )/2}" loading="lazy"></span> für n=4.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Vierdimensionale_Kalotte">Vierdimensionale Kalotte</h3></div>

<p>Eine Kalotte im 4-dimensionalen Raum hat die Mantelfläche

</p>

<msub>

<msup>

</mrow>

</msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle M_{cap^{4}}=r^{3}\pi (2\theta -\sin(2\theta ))}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f23b74d2cf6a602852c4fbdf0a345dfa3b903ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.08ex; height:3.509ex;" alt="{\displaystyle M_{cap^{4}}=r^{3}\pi (2\theta -\sin(2\theta ))}" loading="lazy"></span></dd></dl>

<p>Von Interesse ist hier auch der Rand des Kalottenmantels

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<annotation encoding="application/x-tex">{\displaystyle U_{cap^{4}}=4a^{2}\pi }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b234fc5e9aa063f15f650f504c85420c8f2c48bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.937ex; height:3.509ex;" alt="{\displaystyle U_{cap^{4}}=4a^{2}\pi }" loading="lazy"></span></dd></dl>

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

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

<li><a href="Kugelschicht" title="Kugelschicht">Kugelschicht</a></li>

<li><a href="Kugelring" title="Kugelring">Kugelring</a></li>

<li><a href="Kugelkeil" title="Kugelkeil">Kugelkeil</a></li>

<li><a href="Kreissegment" title="Kreissegment">Kreissegment</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:Spherical_caps?uselang=de"><span lang="en">Commons</span>: Spherical caps</a></span></b> – Sammlung von Bildern, Videos und Audiodateien</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/SphericalCap.html"><i>Spherical Cap</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</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"><a href="Harald_Scheid" title="Harald Scheid">Harald Scheid</a>, Wolfgang Schwarz: <i>Elemente der Geometrie</i> (2009)</span>

</li>

<li id="cite_note-S-Li-2"><span class="mw-cite-backlink"><a href="#cite_ref-S-Li_2-0">↑</a></span> <span class="reference-text">S. Li: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Consice Formulas for the Area and Volume of a Hyperspherical Cap</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Asian Journal of Mathematics and Statistics</cite>. 2011, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>66–70</span> (englisch, <a rel="nofollow" class="external text" href="https://docsdrive.com/pdfs/ansinet/ajms/2011/66-70.pdf">docsdrive.com</a> [PDF]).<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:Kugelsegment&rft.atitle=Consice+Formulas+for+the+Area+and+Volume+of+a+Hyperspherical+Cap&rft.au=S.%26%2332%3BLi&rft.btitle=Asian+Journal+of+Mathematics+and+Statistics&rft.date=2011&rft.genre=book&rft.pages=66-70" style="display:none"> </span></span>

</li>

</ol>

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

<ul><li>Bronstein-Semendjajew: <i>Taschenbuch der Mathematik.</i> Harri-Deutsch-Verlag, 1983, ISBN 3-87144-492-8, S. 252.</li>

<li><i>Kleine Enzyklopädie Mathematik</i>, Harri Deutsch-Verlag, 1977, S. 215.</li></ul></div><div><div class="zim-footer">

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