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The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent and have the same length.

For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the slant height of a lateral face; that is, the shortest distance from apex to base on a given face. For a truncated regular pyramid (a regular pyramid with some of its peak removed by a plane parallel to the base), the apothem is the height of a trapezoidal lateral face.

A triangle has four centers, circumcenter, incenter, centroid, and orthocenter. The center that is used to find the apothem is the incenter.

- $A\; =\; frac\{nsa\}\{2\}\; =\; frac\{pa\}\{2\}.$

An apothem of a regular polygon will always be a radius of the inscribed circle. It is also the distance between any side of the polygon and its center.

The apothem a of a regular n-sided polygon with side length s, or circumradius R, can be found using the following formula:

- $a=frac\{s\}\{2tan(180^circ/n)\}=Rcos(180^circ/n).$

The apothem can also be found by

- $a=frac\{1\}\{2\}stan!left(frac\{90^circ(n-2)\}\{n\}right).$

Both formulae can still be used even if only the perimeter p is known because $s\; =\; frac\{p\}\{n\}.$

- Apothem of a regular polygon With interactive animation
- Apothem of pyramid or truncated pyramid
- Sagitta, Apothem, and Chord by Ed Pegg, Jr., The Wolfram Demonstrations Project.

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Last updated on Wednesday October 08, 2008 at 08:02:53 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday October 08, 2008 at 08:02:53 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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