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Difference between revisions of "Apothem"

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The '''apothem''' of a [[regular polygon]] is the [[line segment]] drawn from the [[center (geometry)|center]] of the [[polygon]] [[perpendicular]] to one of its [[edge]]s. It is also the [[radius]] of the [[incircle | inscribed circle]] of the polygon.
  
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The apothem of a [[regular polygon | regular]] [[polygon]] is the [[line segment]] drawn from the [[center]] of the polygon [[perpendicular]] to one of its [[side]]s. It is also the [[radius]] of the inscribed [[circle]].
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== Formulas ==
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*Given the number of sides, <math>n</math>, and side length, <math>s</math>, the length of the apothem is <math>\frac{s}{2\tan\left(\frac{\pi}{n}\right)}</math>.
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*Given the number of sides, <math>n</math>, and radius of the [[circumcircle | circumscribed circle]], <math>R</math>, the length of the apothem is <math>R\cos\left(\frac{\pi}{n}\right)</math>.
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*Given the length of the apothem, <math>a</math>, and the [[perimeter]], <math>p</math>, of a regular polygon, the [[area]] of the polygon is <math>\frac{ap}{2}</math>, or <math>as</math>, where <math>s</math> is the [[semiperimeter]].
  
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== Formulas ==
 
  
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Given the amount of sides, <math>n</math>, and side length, <math>s</math>, the apothem is <math>\frac{s}{2\tan\left(\frac{\pi}{n}\right)}</math>.
 
  
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Given the amount of sides, <math>n</math>, and radius of the circumscribed circle, <math>R</math>, the apothem is <math>R\cos\left(\frac{\pi}{n}\right)</math>.
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<asy>
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draw((0,sqrt(3))--(1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--cycle);
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draw((1,0)--(2,sqrt(3))--(3,0),gray+linetype("1 2"));
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draw((2,0)--(2,sqrt(3)),gray);
  
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Given the apothem, <math>a</math>, and [[perimeter]], <math>p</math>, the area of the polygon is <math>\frac{ap}{2}</math>, or <math>as</math>, where <math>s</math> is the [[semiperimeter]].
 
  
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</asy>
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{{stub}}
 
[[Category:Definition]]
 
[[Category:Definition]]
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[[Category:Geometry]]

Latest revision as of 17:33, 3 September 2024

The apothem of a regular polygon is the line segment drawn from the center of the polygon perpendicular to one of its edges. It is also the radius of the inscribed circle of the polygon.

Formulas

  • Given the number of sides, $n$, and side length, $s$, the length of the apothem is $\frac{s}{2\tan\left(\frac{\pi}{n}\right)}$.
  • Given the number of sides, $n$, and radius of the circumscribed circle, $R$, the length of the apothem is $R\cos\left(\frac{\pi}{n}\right)$.
  • Given the length of the apothem, $a$, and the perimeter, $p$, of a regular polygon, the area of the polygon is $\frac{ap}{2}$, or $as$, where $s$ is the semiperimeter.


[asy] draw((0,sqrt(3))--(1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--cycle); draw((1,0)--(2,sqrt(3))--(3,0),gray+linetype("1 2")); draw((2,0)--(2,sqrt(3)),gray); [/asy] This article is a stub. Help us out by expanding it.

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