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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Circular segment</span></span>
</h1>
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<p>In <a href="Geometry" title="Geometry">geometry</a>, a <b>circular segment</b> or <b>disk segment</b> (symbol: <span style="font-size:1.5em">⌓</span>) is a region of a <a href="Disk_(mathematics)" title="Disk (mathematics)">disk</a><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> which is "cut off" from the rest of the disk by a straight line. The complete line is known as a <i><a href="Secant_line" title="Secant line">secant</a></i>, and the section inside the disk as a <i><a href="Chord_(geometry)" title="Chord (geometry)">chord</a></i>.<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>
</p><p>More formally, a circular segment is a <a href="Plane_(mathematics)" title="Plane (mathematics)">plane region</a> bounded by a <a href="Circular_arc" title="Circular arc">circular arc</a> (of less than π radians by convention) and the <a href="Circular_chord" class="mw-redirect" title="Circular chord">circular chord</a> connecting its endpoints.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Formulae">Formulae</h2></div>
<p>Let <i>R</i> be the <a href="Radius" title="Radius">radius</a> of the arc which forms part of the perimeter of the segment, <i>θ</i> the <a href="Central_angle" title="Central angle">central angle</a> subtending the arc in <a href="Radian" title="Radian">radians</a>, <i>c</i> the <a href="Chord_length" class="mw-redirect" title="Chord length">chord length</a>, <i>s</i> the <a href="Arc_length" title="Arc length">arc length</a>, <i>h</i> the <a href="Sagitta_(geometry)" title="Sagitta (geometry)">sagitta</a> (<a href="Height#In_mathematics" title="Height">height</a>) of the segment, <i>d</i> the <a href="Apothem" title="Apothem">apothem</a> of the segment, and <i>a</i> the <a href="Area" title="Area">area</a> of the segment.
</p><p>Usually, chord length and height are given or measured, and sometimes the arc length as part of the perimeter, and the unknowns are area and sometimes arc length. These can't be calculated simply from chord length and height, so two intermediate quantities, the radius and central angle are usually calculated first.
</p>
<div class="mw-heading mw-heading3"><h3 id="Radius_and_central_angle">Radius and central angle</h3></div>
<p>The radius is:
</p>
<dl><dd><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={\tfrac {h}{2}}+{\tfrac {c^{2}}{8h}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>h</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mrow>
<mn>8</mn>
<mi>h</mi>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R={\tfrac {h}{2}}+{\tfrac {c^{2}}{8h}}}</annotation>
</semantics>
</math></span><img src="./3e11edd79e5020408a2debc12165442288149b6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.091ex; height:4.176ex;" alt="{\displaystyle R={\tfrac {h}{2}}+{\tfrac {c^{2}}{8h}}}" loading="lazy"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></dd></dl>
<p>The central angle is
</p>
<dl><dd><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 =2\arcsin {\tfrac {c}{2R}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mn>2</mn>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>R</mi>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =2\arcsin {\tfrac {c}{2R}}}</annotation>
</semantics>
</math></span><img src="./59ce4767a1d884d6022ce931a31d136c50c030d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.993ex; height:3.343ex;" alt="{\displaystyle \theta =2\arcsin {\tfrac {c}{2R}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Chord_length_and_height">Chord length and height</h3></div>
<p>The chord length and height can be back-computed from radius and central angle by:
</p><p>The chord length is
</p>
<dl><dd><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 c=2R\sin {\tfrac {\theta }{2}}=R{\sqrt {2(1-\cos \theta )}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo>=</mo>
<mn>2</mn>
<mi>R</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>θ<!-- θ --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo>=</mo>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c=2R\sin {\tfrac {\theta }{2}}=R{\sqrt {2(1-\cos \theta )}}}</annotation>
</semantics>
</math></span><img src="./7e4ee51e729cf8afd0784fd11fbd365578f4b302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.069ex; height:4.843ex;" alt="{\displaystyle c=2R\sin {\tfrac {\theta }{2}}=R{\sqrt {2(1-\cos \theta )}}}" loading="lazy"></span></dd>
<dd><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 c=2{\sqrt {R^{2}-(R-h)^{2}}}=2{\sqrt {2Rh-h^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo>=</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
<mi>R</mi>
<mi>h</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c=2{\sqrt {R^{2}-(R-h)^{2}}}=2{\sqrt {2Rh-h^{2}}}}</annotation>
</semantics>
</math></span><img src="./53842501aacaa0316ec0245aadec1196aff4c488.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:38.141ex; height:4.843ex;" alt="{\displaystyle c=2{\sqrt {R^{2}-(R-h)^{2}}}=2{\sqrt {2Rh-h^{2}}}}" loading="lazy"></span></dd></dl>
<p>The <a href="Sagitta_(geometry)" title="Sagitta (geometry)">sagitta</a> is
</p>
<dl><dd><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=R-{\sqrt {R^{2}-{\frac {c^{2}}{4}}}}=R(1-\cos {\tfrac {\theta }{2}})=R\left(1-{\sqrt {\tfrac {1+\cos \theta }{2}}}\right)={\frac {c}{2}}\tan {\frac {\theta }{4}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>=</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mn>4</mn>
</mfrac>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>θ<!-- θ --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>R</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
</mrow>
<mn>2</mn>
</mfrac>
</mstyle>
</msqrt>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mi>tan</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>θ<!-- θ --></mi>
<mn>4</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h=R-{\sqrt {R^{2}-{\frac {c^{2}}{4}}}}=R(1-\cos {\tfrac {\theta }{2}})=R\left(1-{\sqrt {\tfrac {1+\cos \theta }{2}}}\right)={\frac {c}{2}}\tan {\frac {\theta }{4}}}</annotation>
</semantics>
</math></span><img src="./0616c43a3f1d382df829958519a2b037df8c23c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:67.628ex; height:6.176ex;" alt="{\displaystyle h=R-{\sqrt {R^{2}-{\frac {c^{2}}{4}}}}=R(1-\cos {\tfrac {\theta }{2}})=R\left(1-{\sqrt {\tfrac {1+\cos \theta }{2}}}\right)={\frac {c}{2}}\tan {\frac {\theta }{4}}}" loading="lazy"></span></dd></dl>
<p>The <a href="Apothem" title="Apothem">apothem</a> is
</p>
<dl><dd><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 d=R-h={\sqrt {R^{2}-{\frac {c^{2}}{4}}}}=R\cos {\tfrac {\theta }{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mn>4</mn>
</mfrac>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mi>R</mi>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>θ<!-- θ --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=R-h={\sqrt {R^{2}-{\frac {c^{2}}{4}}}}=R\cos {\tfrac {\theta }{2}}}</annotation>
</semantics>
</math></span><img src="./7bd73f6fe6935e72e071a422bbb247b9651c3f72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.642ex; height:6.176ex;" alt="{\displaystyle d=R-h={\sqrt {R^{2}-{\frac {c^{2}}{4}}}}=R\cos {\tfrac {\theta }{2}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Arc_length_and_area">Arc length and area</h3></div>
<p>The arc length, from the familiar geometry of a circle, is
</p>
<dl><dd><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 s={\theta }R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s={\theta }R}</annotation>
</semantics>
</math></span><img src="./de6ab783fad91f23c8749729841049140409de06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.043ex; height:2.176ex;" alt="{\displaystyle s={\theta }R}" loading="lazy"></span></dd></dl>
<p>The area <i>a</i> of the circular segment is equal to the area of the <a href="Circular_sector" title="Circular sector">circular sector</a> minus the area of the triangular portion (using the double angle formula to get an equation 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 \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./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>):
</p>
<dl><dd><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={\tfrac {R^{2}}{2}}\left(\theta -\sin \theta \right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a={\tfrac {R^{2}}{2}}\left(\theta -\sin \theta \right)}</annotation>
</semantics>
</math></span><img src="./a83ef0a7a66cd99d8d5a24e72c5c806d80a23010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.704ex; height:4.009ex;" alt="{\displaystyle a={\tfrac {R^{2}}{2}}\left(\theta -\sin \theta \right)}" loading="lazy"></span></dd></dl>
<p>In terms of <span class="texhtml"><i>c</i></span> and <span class="texhtml"><i>R</i></span>,
</p>
<dl><dd><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={\tfrac {R^{2}}{2}}\left(2\arcsin {\tfrac {c}{2R}}-\sin \left(2\arcsin {\tfrac {c}{2R}}\right)\right)=R^{2}\left(\arcsin {\frac {c}{2R}}-{\frac {c}{2R}}{\sqrt {1-\left({\frac {c}{2R}}\right)^{2}}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>R</mi>
</mrow>
</mfrac>
</mstyle>
</mrow>
<mo>−<!-- − --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>R</mi>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>R</mi>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>R</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>R</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a={\tfrac {R^{2}}{2}}\left(2\arcsin {\tfrac {c}{2R}}-\sin \left(2\arcsin {\tfrac {c}{2R}}\right)\right)=R^{2}\left(\arcsin {\frac {c}{2R}}-{\frac {c}{2R}}{\sqrt {1-\left({\frac {c}{2R}}\right)^{2}}}\right)}</annotation>
</semantics>
</math></span><img src="./922a02b162b75c5c948c26142d17cfdd89274cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:80.844ex; height:6.343ex;" alt="{\displaystyle a={\tfrac {R^{2}}{2}}\left(2\arcsin {\tfrac {c}{2R}}-\sin \left(2\arcsin {\tfrac {c}{2R}}\right)\right)=R^{2}\left(\arcsin {\frac {c}{2R}}-{\frac {c}{2R}}{\sqrt {1-\left({\frac {c}{2R}}\right)^{2}}}\right)}" loading="lazy"></span></dd></dl>
<p>In terms of <span class="texhtml"><i>R</i></span> and <span class="texhtml"><i>h</i></span>,
</p>
<dl><dd><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^{2}\arccos \left(1-{\frac {h}{R}}\right)-\left(R-h\right){\sqrt {R^{2}-\left(R-h\right)^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>arccos</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mi>R</mi>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=R^{2}\arccos \left(1-{\frac {h}{R}}\right)-\left(R-h\right){\sqrt {R^{2}-\left(R-h\right)^{2}}}}</annotation>
</semantics>
</math></span><img src="./dd186f085a3d32f4c6b141344fb38b569e122868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.545ex; height:6.176ex;" alt="{\displaystyle a=R^{2}\arccos \left(1-{\frac {h}{R}}\right)-\left(R-h\right){\sqrt {R^{2}-\left(R-h\right)^{2}}}}" loading="lazy"></span></dd></dl>
<p>In terms of <span class="texhtml"><i>c</i></span> and <span class="texhtml"><i>h</i></span>,
</p>
<dl><dd><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=\left({\frac {c^{2}+4h^{2}}{8h}}\right)^{2}\arccos \left({\frac {c^{2}-4h^{2}}{c^{2}+4h^{2}}}\right)-{\frac {c}{16h}}(c^{2}-4h^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>4</mn>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>8</mn>
<mi>h</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>arccos</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>4</mn>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>4</mn>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>c</mi>
<mrow>
<mn>16</mn>
<mi>h</mi>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>4</mn>
<msup>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=\left({\frac {c^{2}+4h^{2}}{8h}}\right)^{2}\arccos \left({\frac {c^{2}-4h^{2}}{c^{2}+4h^{2}}}\right)-{\frac {c}{16h}}(c^{2}-4h^{2})}</annotation>
</semantics>
</math></span><img src="./81d0db10738a5499f56c01e253a4c8b896e93e4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.023ex; height:6.676ex;" alt="{\displaystyle a=\left({\frac {c^{2}+4h^{2}}{8h}}\right)^{2}\arccos \left({\frac {c^{2}-4h^{2}}{c^{2}+4h^{2}}}\right)-{\frac {c}{16h}}(c^{2}-4h^{2})}" loading="lazy"></span></dd></dl>
<p>What can be stated is that as the central angle gets smaller (or alternately the radius gets larger), the area <i>a</i> rapidly and asymptotically approaches <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 {\tfrac {2}{3}}c\cdot h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mstyle>
</mrow>
<mi>c</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}c\cdot h}</annotation>
</semantics>
</math></span><img src="./d5d654400169d6b8c8ab0fd7288acc5b61da44be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.683ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}c\cdot h}" loading="lazy"></span>. If <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 \ll 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>≪<!-- ≪ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta \ll 1}</annotation>
</semantics>
</math></span><img src="./0d879842f3f2778cca6422dc8364e3994fe95430.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.867ex; height:2.176ex;" alt="{\displaystyle \theta \ll 1}" 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 a={\tfrac {2}{3}}c\cdot h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mstyle>
</mrow>
<mi>c</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a={\tfrac {2}{3}}c\cdot h}</annotation>
</semantics>
</math></span><img src="./7726a5a599bc4448d4be3ae5d8a8d6e6da9e3ec5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.011ex; height:3.676ex;" alt="{\displaystyle a={\tfrac {2}{3}}c\cdot h}" loading="lazy"></span> is a substantially good approximation.
</p><p>If <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 c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
</semantics>
</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> is held constant, and the radius is allowed to vary, then we have<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 {\frac {\partial a}{\partial s}}=R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>a</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>s</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial a}{\partial s}}=R}</annotation>
</semantics>
</math></span></span>
</p><p>As the central angle approaches π, the area of the segment is converging to the area of a <a href="Semicircle" title="Semicircle">semicircle</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 {\tfrac {\pi R^{2}}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi R^{2}}{2}}}</annotation>
</semantics>
</math></span><img src="./5a71df4e5223f106fb1999726a97743749165661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.857ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\pi R^{2}}{2}}}" loading="lazy"></span>, so a good approximation is a delta offset from the latter area:
</p>
<dl><dd><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\approx {\tfrac {\pi R^{2}}{2}}-(R+{\tfrac {c}{2}})(R-h)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>≈<!-- ≈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\approx {\tfrac {\pi R^{2}}{2}}-(R+{\tfrac {c}{2}})(R-h)}</annotation>
</semantics>
</math></span><img src="./957a998247bc49cb21a4242817a2991d65c1206b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.85ex; height:4.009ex;" alt="{\displaystyle a\approx {\tfrac {\pi R^{2}}{2}}-(R+{\tfrac {c}{2}})(R-h)}" loading="lazy"></span> for h>.75<i>R</i></dd></dl>
<p>As an example, the area is one quarter the circle when <i>θ</i> ~ 2.31 radians (132.3°) corresponding to a height of ~59.6% and a chord length of ~183% of the radius.
</p>
<div class="mw-heading mw-heading3"><h3 id="Other_properties">Other properties</h3></div>
<p>The perimeter <i>p</i> is the arclength plus the chord length:
</p>
<dl><dd><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 p=c+s=c+\theta R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>c</mi>
<mo>+</mo>
<mi>s</mi>
<mo>=</mo>
<mi>c</mi>
<mo>+</mo>
<mi>θ<!-- θ --></mi>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=c+s=c+\theta R}</annotation>
</semantics>
</math></span><img src="./bd4e360bc00368ba31f6d8da7dc637aeb77e8f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:19.095ex; height:2.509ex;" alt="{\displaystyle p=c+s=c+\theta R}" loading="lazy"></span></dd></dl>
<p>Proportion of the whole area of the circle:
</p>
<dl><dd><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 {\frac {a}{A}}={\frac {\theta -\sin \theta }{2\pi }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>a</mi>
<mi>A</mi>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{A}}={\frac {\theta -\sin \theta }{2\pi }}}</annotation>
</semantics>
</math></span><img src="./6337589e874396a55dbe8b3273818203ddd41603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.778ex; height:5.509ex;" alt="{\displaystyle {\frac {a}{A}}={\frac {\theta -\sin \theta }{2\pi }}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>The area formula can be used in calculating the volume of a partially-filled cylindrical tank lying horizontally.
</p><p>In the design of windows or doors with rounded tops, <i>c</i> and <i>h</i> may be the only known values and can be used to calculate <i>R</i> for the draftsman's compass setting.
</p><p>One can reconstruct the full dimensions of a complete circular object from fragments by measuring the arc length and the chord length of the fragment.
</p><p>To check hole positions on a circular pattern. Especially useful for quality checking on machined products.
</p><p>For calculating the area or locating the centroid of a planar shape that contains circular segments.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Chord_(geometry)" title="Chord (geometry)">Chord (geometry)</a></li>
<li><a href="Spherical_cap" title="Spherical cap">Spherical cap</a></li>
<li><a href="Circular_sector" title="Circular sector">Circular sector</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Mathematics distinguishes when necessary between the words <i>circle</i> and <i>disk</i>: a disk is a plane area having a circle as its boundary, while a circle is the closed curve forming the boundary itself.</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">These terms refer to a line which intersects a curve. In this case, the curve is the circle forming the disk's boundary.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">The fundamental relationship between <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</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>, <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 c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
</semantics>
</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle 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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</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> derivable directly from the Pythagorean theorem among <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</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>, <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 c/2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c/2}</annotation>
</semantics>
</math></span><img src="./31dc515ca5e197ffb2fa35017b3d96b778b5b6fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.332ex; height:2.843ex;" alt="{\displaystyle c/2}" 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 R-h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R-h}</annotation>
</semantics>
</math></span><img src="./807eb1e6acd1ecd785f9e7e5dcd19651900c38fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.943ex; height:2.343ex;" alt="{\displaystyle R-h}" loading="lazy"></span> as components of a right triangle 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^{2}=({\tfrac {c}{2}})^{2}+(R-h)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>c</mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R^{2}=({\tfrac {c}{2}})^{2}+(R-h)^{2}}</annotation>
</semantics>
</math></span><img src="./fa89076f3757e19323234dac0c18fafface57939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.086ex; height:3.509ex;" alt="{\displaystyle R^{2}=({\tfrac {c}{2}})^{2}+(R-h)^{2}}" loading="lazy"></span> which may be solved 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 R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</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>, <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 c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
</semantics>
</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>, or <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</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 required.</span>
</li>
</ol></div></div>
<ul><li><span class="citation mathworld" id="Reference-Mathworld-Circular_segment"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CircularSegment.html">"Circular segment"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/segment.html">Definition of a circular segment</a> With interactive animation</li>
<li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/segmentarea.html">Formulae for area of a circular segment</a> With interactive animation</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-07-08" href="https://en.wikipedia.org/wiki/?title=Circular_segment&oldid=1299525835">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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