apothem

The word "apothem" can also refer to the length of that line segment.

The area of a regular polygon is half its perimeter times the apothem.

The inradius of a regular polygon is also called apothem.

This radius is also termed its apothem and is often represented as a.

The apothem of a regular polygon can be found multiple ways, of which two are described here.

An apothem of a regular polygon will always be a radius of the inscribed circle.

The area can also be found by the formulas and , where a is the apothem and p is the perimeter.

The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides.

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

As the number of sides of the regular polygon increases, it becomes identical to a circle, and the apothem becomes identical to the radius.

Specifically, it equals n times the apothem, where n is the number of sides and the apothem is the distance from the center to a side.

Sagitta, Apothem, and Chord by Ed Pegg, Jr., The Wolfram Demonstrations Project.

A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a "golden pyramid".

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.

For an equilateral triangle, the apothem is equivalent to the line segment from the midpoint of a side to any of the triangle's centers, since an equilateral triangle's centers coincide as a consequence of the definition.

For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem (the apothem being the distance from the center to any side).

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.

The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem.

This formula can be derived by partitioning the n-sided polygon into n congruent isosceles triangles, and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height.

Note that, given a regular pentagonal dodecahedron of edge length one, r is the radius of a circumscribing sphere about a cube of edge length φ, and r is the apothem of a regular pentagon of edge length φ.