Editing (section) Volume 0 You are not logged in. The rich text editor does not work with JavaScript switched off. Please either enable it in your browser options, or visit your preferences to switch to the old MediaWiki editor <h2 data-rte-spaces-before="1" data-rte-spaces-after="1"> Volume formula derivations </h2> <p data-rte-fromparser="true" data-rte-filler="true"></p><h3 data-rte-spaces-before="1" data-rte-spaces-after="1"> Sphere </h3> <p data-rte-fromparser="true" data-rte-empty-lines-before="1">The volume of a <span class="placeholder placeholder-double-brackets" data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22double-brackets%22%2C%22wikitext%22%3A%22%7B%7BDirected%20link%7Csphere%7D%7D%22%2C%22lineStart%22%3A%22%22%2C%22title%22%3A%22Directed%20link%22%2C%22placeholder%22%3A1%7D" contenteditable="false">​<a data-rte-meta="%7B%22type%22%3A%22internal%22%2C%22wikitextIdx%22%3Anull%2C%22text%22%3A%22sphere%22%2C%22link%22%3A%22Wikipedia%3Asphere%22%2C%22wasblank%22%3Afalse%2C%22noforce%22%3Atrue%7D" data-rte-instance="6456-4676905545f023c90c6920" href="http://en.wikipedia.org/wiki/sphere" class="extiw" title="wikipedia:sphere">sphere</a>​</span> is the <span class="placeholder placeholder-double-brackets" data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22double-brackets%22%2C%22wikitext%22%3A%22%7B%7BDirected%20link%7Cintegral%7D%7D%22%2C%22lineStart%22%3A%22%22%2C%22title%22%3A%22Directed%20link%22%2C%22placeholder%22%3A1%7D" contenteditable="false">​<a data-rte-meta="%7B%22type%22%3A%22internal%22%2C%22wikitextIdx%22%3Anull%2C%22text%22%3A%22integral%22%2C%22link%22%3A%22Wikipedia%3Aintegral%22%2C%22wasblank%22%3Afalse%2C%22noforce%22%3Atrue%7D" data-rte-instance="6456-4676905545f023c90c6920" href="http://en.wikipedia.org/wiki/integral" class="extiw" title="wikipedia:integral">integral</a>​</span> of infinitesimal circular slabs of thickness <i>dx</i>. <!-- RTE::{"spaces":0,"type":"LINE_BREAK"} -->The calculation for the volume of a sphere with center 0 and radius <i>r</i> is as follows. </p> The surface area of the circular slab is <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%5C%5Cpi%20r%5E2%20%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \pi r^2 $</span>​</div>. <p data-rte-fromparser="true" data-rte-empty-lines-before="1">The radius of the circular slabs, defined such that the x-axis cuts perpendicularly through them, is; </p> <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3Ey%20%3D%20%5C%5Csqrt%7Br%5E2-x%5E2%7D%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ y = \sqrt{r^2-x^2} $</span>​</div> <p data-rte-fromparser="true" data-rte-empty-lines-before="1">or </p> <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3Ez%20%3D%20%5C%5Csqrt%7Br%5E2-x%5E2%7D%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ z = \sqrt{r^2-x^2} $</span>​</div> <p data-rte-fromparser="true" data-rte-empty-lines-before="1">where y or z can be taken to represent the radius of a slab at a particular x value. </p> Using y as the slab radius, the volume of the sphere can be calculated as <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%20%5C%5Cint_%7B-r%7D%5Er%20%5C%5Cpi%20y%5E2%20%5C%5C%2Cdx%20%3D%20%5C%5Cint_%7B-r%7D%5Er%20%5C%5Cpi%28r%5E2%20-%20x%5E2%29%20%5C%5C%2Cdx.%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \int_{-r}^r \pi y^2 \,dx = \int_{-r}^r \pi(r^2 - x^2) \,dx. $</span>​</div> Now <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%5C%5Cint_%7B-r%7D%5Er%20%5C%5Cpi%20r%5E2%5C%5C%2Cdx%20-%20%5C%5Cint_%7B-r%7D%5Er%20%5C%5Cpi%20x%5E2%5C%5C%2Cdx%20%3D%20%5C%5Cpi%20%28r%5E3%20%2B%20r%5E3%29%20-%20%5C%5Cfrac%7B%5C%5Cpi%7D%7B3%7D%28r%5E3%20%2B%20r%5E3%29%20%3D%20%202%5C%5Cpi%20r%5E3%20-%20%5C%5Cfrac%7B2%5C%5Cpi%20r%5E3%7D%7B3%7D.%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \int_{-r}^r \pi r^2\,dx - \int_{-r}^r \pi x^2\,dx = \pi (r^3 + r^3) - \frac{\pi}{3}(r^3 + r^3) = 2\pi r^3 - \frac{2\pi r^3}{3}. $</span>​</div> Combining yields gives <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3EV%20%3D%20%5C%5Cfrac%7B4%7D%7B3%7D%5C%5Cpi%20r%5E3.%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ V = \frac{4}{3}\pi r^3. $</span>​</div> This formula can be derived more quickly using the formula for the sphere's <span class="placeholder placeholder-double-brackets" data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22double-brackets%22%2C%22wikitext%22%3A%22%7B%7BDirected%20link%7Csurface%20area%7D%7D%22%2C%22lineStart%22%3A%22%22%2C%22title%22%3A%22Directed%20link%22%2C%22placeholder%22%3A1%7D" contenteditable="false">​<a data-rte-meta="%7B%22type%22%3A%22internal%22%2C%22wikitextIdx%22%3Anull%2C%22text%22%3A%22surface%20area%22%2C%22link%22%3A%22Wikipedia%3Asurface%20area%22%2C%22wasblank%22%3Afalse%2C%22noforce%22%3Atrue%7D" data-rte-instance="6456-4676905545f023c90c6920" href="http://en.wikipedia.org/wiki/surface_area" class="extiw" title="wikipedia:surface area">surface area</a>​</span>, which is <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E4%5C%5Cpi%20r%5E2%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ 4\pi r^2 $</span>​</div>. <p data-rte-fromparser="true">The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to </p> <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%20%5C%5Cint_0%5Er%204%5C%5Cpi%20u%5E2%20%5C%5C%2Cdu%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \int_0^r 4\pi u^2 \,du $</span>​</div> = <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%20%20%5C%5Cfrac%7B4%7D%7B3%7D%5C%5Cpi%20r%5E3.%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false"><span class="tex" dir="ltr">$ \frac{4}{3}\pi r^3. $</span>​</div> <h3 data-rte-spaces-before="1" data-rte-spaces-after="1" data-rte-empty-lines-before="1"> Cone </h3> <p data-rte-fromparser="true">The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies cones as well. But for an explanation using calculus: </p><p data-rte-fromparser="true" data-rte-empty-lines-before="1">The volume of a <span class="placeholder placeholder-double-brackets" data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22double-brackets%22%2C%22wikitext%22%3A%22%7B%7BDirected%20link%7CCone%20%28geometry%29%7Ccone%7D%7D%22%2C%22lineStart%22%3A%22%22%2C%22title%22%3A%22Directed%20link%22%2C%22placeholder%22%3A1%7D" contenteditable="false">​<a data-rte-meta="%7B%22type%22%3A%22internal%22%2C%22wikitextIdx%22%3Anull%2C%22text%22%3A%22cone%22%2C%22link%22%3A%22Wikipedia%3ACone%20%28geometry%29%22%2C%22wasblank%22%3Afalse%2C%22noforce%22%3Atrue%7D" data-rte-instance="6456-4676905545f023c90c6920" href="http://en.wikipedia.org/wiki/Cone_(geometry)" class="extiw" title="wikipedia:Cone (geometry)">cone</a>​</span> is the <span class="placeholder placeholder-double-brackets" data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22double-brackets%22%2C%22wikitext%22%3A%22%7B%7BDirected%20link%7Cintegral%7D%7D%22%2C%22lineStart%22%3A%22%22%2C%22title%22%3A%22Directed%20link%22%2C%22placeholder%22%3A1%7D" contenteditable="false">​<a data-rte-meta="%7B%22type%22%3A%22internal%22%2C%22wikitextIdx%22%3Anull%2C%22text%22%3A%22integral%22%2C%22link%22%3A%22Wikipedia%3Aintegral%22%2C%22wasblank%22%3Afalse%2C%22noforce%22%3Atrue%7D" data-rte-instance="6456-4676905545f023c90c6920" href="http://en.wikipedia.org/wiki/integral" class="extiw" title="wikipedia:integral">integral</a>​</span> of infinitesimal circular slabs of thickness <i>dx</i>. <!-- RTE::{"spaces":0,"type":"LINE_BREAK"} -->The calculation for the volume of a cone of height <i>h</i>, whose base is centered at (0,0,0) with radius <i>r</i>, is as follows. </p> The radius of each circular slab is <i>r</i> if <i>x</i> = 0 and 0 if <i>x</i> = <i>h</i>, and varying linearly in between—that is, <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3Er%5C%5Cfrac%7B%28h-x%29%7D%7Bh%7D.%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ r\frac{(h-x)}{h}. $</span>​</div> The surface area of the circular slab is then <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%20%5C%5Cpi%20%5C%5Cleft%28r%5C%5Cfrac%7B%28h-x%29%7D%7Bh%7D%5C%5Cright%29%5E2%20%3D%20%20%5C%5Cpi%20r%5E2%5C%5Cfrac%7B%28h-x%29%5E2%7D%7Bh%5E2%7D.%20%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \pi \left(r\frac{(h-x)}{h}\right)^2 = \pi r^2\frac{(h-x)^2}{h^2}. $</span>​</div> The volume of the cone can then be calculated as <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%20%5C%5Cint_%7B0%7D%5Eh%20%5C%5Cpi%20r%5E2%5C%5Cfrac%7B%28h-x%29%5E2%7D%7Bh%5E2%7D%20dx%2C%20%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \int_{0}^h \pi r^2\frac{(h-x)^2}{h^2} dx, $</span>​</div> and after extraction of the constants: <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%5C%5Cfrac%7B%5C%5Cpi%20r%5E2%7D%7Bh%5E2%7D%20%5C%5Cint_%7B0%7D%5Eh%20%28h-x%29%5E2%20dx%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \frac{\pi r^2}{h^2} \int_{0}^h (h-x)^2 dx $</span>​</div> Integrating gives us <div data-rte-instance="6456-4676905545f023c90c6920" data-rte-meta="%7B%22type%22%3A%22ext%22%2C%22wikitext%22%3A%22%3Cmath%3E%5C%5Cfrac%7B%5C%5Cpi%20r%5E2%7D%7Bh%5E2%7D%5C%5Cleft%28%5C%5Cfrac%7Bh%5E3%7D%7B3%7D%5C%5Cright%29%20%3D%20%5C%5Cfrac%7B1%7D%7B3%7D%5C%5Cpi%20r%5E2%20h.%3C%5C%2Fmath%3E%22%2C%22lineStart%22%3A%22%22%2C%22placeholder%22%3A1%2C%22extName%22%3A%22math%22%7D" class="placeholder placeholder-ext" contenteditable="false" data-rte-empty-lines-before="1"><span class="tex" dir="ltr">$ \frac{\pi r^2}{h^2}\left(\frac{h^3}{3}\right) = \frac{1}{3}\pi r^2 h. $</span>​</div> <p /> <!-- Saved in parser cache with key units:rte-parser-cache:2356 --> Loading editor Below are some commonly used wiki markup codes. 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