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<title>Napkin ring problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Napkin ring problem</span></span>

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<p>In <a href="Geometry" title="Geometry">geometry</a>, the <b>napkin-ring problem</b> involves finding the volume of a "band" of specified height around a <a href="Sphere" title="Sphere">sphere</a>, i.e. the part that remains after a hole in the shape of a circular cylinder is drilled through the center of the sphere. It is a counterintuitive fact that this volume does not depend on the original sphere's <a href="Radius" title="Radius">radius</a> but only on the resulting band's height.

</p><p>The problem is so called because after removing a cylinder from the sphere, the remaining band resembles the shape of a <a href="Napkin_ring" title="Napkin ring">napkin ring</a>.

</p>

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

<p>Suppose that the axis of a <a href="Right_circular_cylinder" title="Right circular cylinder">right circular cylinder</a> passes through the center of a sphere of radius <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> and that <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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> represents the height (defined as the distance in a direction <a href="Parallel_(geometry)" title="Parallel (geometry)">parallel</a> to the axis) of the part of the cylinder that is inside the sphere. The "band" is the part of the sphere that is outside the cylinder. The volume of the band depends on <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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> but not on <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>:

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<msup>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {\pi h^{3}}{6}}.}</annotation>

</semantics>

</math></span></span>

</p><p>As the radius <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> of the sphere shrinks, the diameter of the cylinder must also shrink in order that <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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> can remain the same. The band gets thicker, and this would increase its volume. But it also gets shorter in circumference, and this would decrease its volume. The two effects exactly cancel each other out. In the extreme case of the smallest possible sphere, the cylinder vanishes (its radius becomes zero) and the height <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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> equals the diameter of the sphere. In this case the volume of the band is the <a href="Volume_of_a_sphere" class="mw-redirect" title="Volume of a sphere">volume of the whole sphere</a>, which matches the formula given above.

</p><p>An early study of this problem was written by 17th-century <a href="Japanese_mathematics" title="Japanese mathematics">Japanese mathematician</a> <a href="Seki_K%C5%8Dwa" class="mw-redirect" title="Seki Kōwa">Seki Kōwa</a>. According to <a href="#CITEREFSmithMikami1914">Smith & Mikami (1914)</a>, Seki called this solid an arc-ring, or in <a href="Japanese_language" title="Japanese language">Japanese</a> <i>kokan</i> or <i>kokwan</i>.<sup id="cite_ref-smith-mikami_1-0" class="reference"><a href="#cite_note-smith-mikami-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>Suppose the radius of the sphere is <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> and the length of the cylinder (or the tunnel) is <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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>.

</p><p>By the <a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, the radius of the cylinder is

</p>

<msqrt>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {R^{2}-\left({\frac {h}{2}}\right)^{2}}},\qquad \qquad (1)}</annotation>

</semantics>

</math></span></span>

and the radius of the horizontal cross-section of the sphere at height <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> above the "equator" is

<msqrt>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {R^{2}-y^{2}}}.\qquad \qquad (2)}</annotation>

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</math></span></span>

</p><p>The <a href="Cross-section_(geometry)" class="mw-redirect" title="Cross-section (geometry)">cross-section</a> of the band with the plane at height <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> is the region inside the larger circle of radius given by (2) and outside the smaller circle of radius given by (1). The cross-section's area is therefore the area of the larger circle minus the area of the smaller circle:

<mtr>

<mtd>

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<mtext>larger radius</mtext>

</mrow>

<msup>

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</msup>

<mtext>smaller radius</mtext>

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<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

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</msup>

</msqrt>

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</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<msqrt>

<msup>

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</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

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</mrow>

</msup>

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</msqrt>

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</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

<mrow>

<mfrac>

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</mrow>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{}\quad \pi ({\text{larger radius}})^{2}-\pi ({\text{smaller radius}})^{2}\\&=\pi \left({\sqrt {R^{2}-y^{2}}}\right)^{2}-\pi \left({\sqrt {R^{2}-\left({\frac {h}{2}}\right)^{2}\,{}}}\,\right)^{2}=\pi \left(\left({\frac {h}{2}}\right)^{2}-y^{2}\right).\end{aligned}}}</annotation>

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</math></span></span>

</p><p>The radius <i>R</i> does not appear in the last quantity. Therefore, the area of the horizontal cross-section at height <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> does not depend on <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>, as long as <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\leq {\tfrac {h}{2}}\leq R}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\leq {\tfrac {h}{2}}\leq R}</annotation>

</semantics>

</math></span><img src="./4d22ca040141e47db2a055194c7c0a8b3c53a233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.899ex; height:3.676ex;" alt="{\displaystyle y\leq {\tfrac {h}{2}}\leq R}" loading="lazy"></span>. The volume of the band is

</p>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mtext>area of cross-section at height </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{-h/2}^{h/2}({\text{area of cross-section at height }}y)\,dy,}</annotation>

</semantics>

</math></span><img src="./901a65cdf400ebf7959b4f9c424e78704ad19dff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.91ex; height:6.676ex;" alt="{\displaystyle \int _{-h/2}^{h/2}({\text{area of cross-section at height }}y)\,dy,}" loading="lazy"></span></dd></dl>

<p>and that does not depend on <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>.

</p><p>This is an application of <a href="Cavalieri's_principle" title="Cavalieri's principle">Cavalieri's principle</a>: volumes with equal-sized corresponding cross-sections are equal. Indeed, the area of the cross-section is the same as that of the corresponding cross-section of a sphere of radius <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/2}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./fac4e3b62775476fc9d9eab607fc71093d6f7328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.664ex; height:2.843ex;" alt="{\displaystyle h/2}" loading="lazy"></span>, which has volume

<mfrac>

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<msup>

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</msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {4}{3}}\pi \left({\frac {h}{2}}\right)^{3}={\frac {\pi h^{3}}{6}}.}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Another_Derivation">Another Derivation</h2></div>

<p>We can also find the napkin ring's volume <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="./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> using previous results.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Specifically, volume <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="./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> must equal the original sphere's volume <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\pi R^{3}/3}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi R^{3}/3}</annotation>

</semantics>

</math></span><img src="./b39e7cc16420779afd147e0658aa02fd1906b4e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.638ex; height:3.176ex;" alt="{\displaystyle 4\pi R^{3}/3}" loading="lazy"></span> <i>minus</i> the cylinder's volume <i>minus</i> the volume of two <a href="Spherical_cap" title="Spherical cap">spherical caps</a>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}(R,h,r_{c},h_{s})=\left\{{\frac {4}{3}}\pi R^{3}\right\}-\left\{\pi (r_{c})^{2}\,h\right\}-2\left\{{\frac {\pi }{3}}(h_{s})^{2}(3R-h_{s})\right\}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{n}(R,h,r_{c},h_{s})=\left\{{\frac {4}{3}}\pi R^{3}\right\}-\left\{\pi (r_{c})^{2}\,h\right\}-2\left\{{\frac {\pi }{3}}(h_{s})^{2}(3R-h_{s})\right\}}</annotation>

</semantics>

</math></span></span>

</p><p>In the above, the cylinder volume uses its radius <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_{c}}">

<msub>

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

</semantics>

</math></span><img src="./5e41dc9dcda9fd0c224d94555361f02b73aacd2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.993ex; height:2.009ex;" alt="{\displaystyle r_{c}}" loading="lazy"></span> which can be written in terms of <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> and <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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> as shown in (1)

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{c}={\sqrt {R^{2}-\left({\frac {h}{2}}\right)^{2}}}}">

<msub>

</mrow>

</msub>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{c}={\sqrt {R^{2}-\left({\frac {h}{2}}\right)^{2}}}}</annotation>

</semantics>

</math></span></span>

</p><p>The spherical cap volume used in <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="./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> uses the cap's height <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_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b6d75beabd0038674f5545e00f8fa5c4f38fb63b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:2.509ex;" alt="{\displaystyle h_{s}}" loading="lazy"></span>. This is found by knowing the height of the sphere <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 2R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c5e012ed1fbb85fd15e40e08a4f375e37650c4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.926ex; height:2.176ex;" alt="{\displaystyle 2R}" loading="lazy"></span> also equals the cylinder's height <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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> plus two spherical cap heights

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2R=h+2h_{s}\qquad \implies \qquad h_{s}=R-{\frac {h}{2}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2R=h+2h_{s}\qquad \implies \qquad h_{s}=R-{\frac {h}{2}}}</annotation>

</semantics>

</math></span></span>

</p><p>Substituting <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_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5e41dc9dcda9fd0c224d94555361f02b73aacd2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.993ex; height:2.009ex;" alt="{\displaystyle r_{c}}" loading="lazy"></span> and <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_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b6d75beabd0038674f5545e00f8fa5c4f38fb63b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:2.509ex;" alt="{\displaystyle h_{s}}" loading="lazy"></span> into the expression above for <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="./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> one finds all terms containing <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> cancel and one gets

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}={\frac {\pi }{6}}h^{3}}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{n}={\frac {\pi }{6}}h^{3}}</annotation>

</semantics>

</math></span></span>

</p>

<ul><li><a href="Visual_calculus" title="Visual calculus">Visual calculus</a>, an intuitive way to solve this type of problem, originally applied to finding the area of an <a href="Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a>, given only its <a href="Chord_(geometry)" title="Chord (geometry)">chord</a> length</li>

<li><a href="String_girdling_Earth" title="String girdling Earth">String girdling Earth</a>, another problem where the radius of a sphere or circle is counter-intuitively irrelevant</li></ul>

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

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</style><cite id="CITEREFSmithMikami1914" class="citation cs2"><a href="David_Eugene_Smith" title="David Eugene Smith">Smith, David E.</a>; <a href="Yoshio_Mikami" title="Yoshio Mikami">Mikami, Yoshio</a> (1914), <a rel="nofollow" class="external text" href="https://archive.org/details/ahistoryjapanes01mikagoog/page/n132/mode/2up"><i>A History of Japanese Mathematics</i></a>, Open Court, pp. <span class="nowrap">121–</span>123, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-43482-6</bdi></cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{citation}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span>.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">See Devlin, Keith (2008), The Napkin Ring Problem under Further Reading.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFDevlin2008" class="citation cs2"><a href="Keith_Devlin" title="Keith Devlin">Devlin, Keith</a> (2008), <a rel="nofollow" class="external text" href="http://www.maa.org/devlin/devlin_04_08.html"><i>The Napkin Ring Problem</i></a>, <a href="Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080430202345/http://www.maa.org/devlin/devlin_04_08.html">archived</a> from the original on 30 April 2008<span class="reference-accessdate">, retrieved <span class="nowrap">25 February</span> 2009</span></cite></li>

<li><cite id="CITEREFDevlin2008" class="citation cs2"><a href="Keith_Devlin" title="Keith Devlin">Devlin, Keith</a> (2008), <a rel="nofollow" class="external text" href="http://www.maa.org/devlin/devlin_05_08.html"><i>Lockhart's Lament</i></a>, <a href="Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080510145548/http://www.maa.org/devlin/devlin_05_08.html">archived</a> from the original on 10 May 2008<span class="reference-accessdate">, retrieved <span class="nowrap">25 February</span> 2009</span></cite></li>

<li><cite id="CITEREFGardner1994" class="citation cs2"><a href="Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (1994), "Hole in the Sphere", <i>My best mathematical and logic puzzles</i>, <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, p. 8</cite></li>

<li><cite id="CITEREFJones1912" class="citation cs2">Jones, Samuel I. (1912), <i>Mathematical Wrinkles for Teachers and Private Learners</i>, Norwood, MA: J. B. Cushing Co.</cite> Problem 132 asks for the volume of a sphere with a cylindrical hole drilled through it, but does not note the invariance of the problem under changes of radius.</li>

<li><cite id="CITEREFLevi2009" class="citation cs2">Levi, Mark (2009), "6.3 How Much Gold Is in a Wedding Ring?", <i>The Mathematical Mechanic: Using Physical Reasoning to Solve Problems</i>, Princeton University Press, pp. <span class="nowrap">102–</span>104, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-14020-9</bdi></cite>. Levi argues that the volume depends only on the height of the hole based on the fact that the ring can be swept out by a half-disk with the height as its diameter.</li>

<li><cite id="CITEREFLines1965" class="citation cs2">Lines, L. (1965), <i>Solid geometry: With Chapters on Space-lattices, Sphere-packs and Crystals</i>, Dover</cite>. Reprint of 1935 edition. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a "napkin ring" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole.</li>

<li><cite id="CITEREFPólya1990" class="citation cs2"><a href="George_P%C3%B3lya" title="George Pólya">Pólya, George</a> (1990), <i><a href="Mathematics_and_plausible_reasoning" class="mw-redirect" title="Mathematics and plausible reasoning">Mathematics and Plausible Reasoning</a>, Vol. I: Induction and Analogy in Mathematics</i>, Princeton University Press, pp. <span class="nowrap">191–</span>192</cite>. Reprint of 1954 edition.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<ul><li><span class="citation mathworld" id="Reference-Mathworld-Spherical_Ring"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SphericalRing.html">"Spherical Ring"</a>, <i><a href="MathWorld" title="MathWorld">MathWorld</a></i></cite></span></li></ul></div><div><div class="zim-footer">

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