Apothem

The apothem (sometimes abbreviated as apo[1]) 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 and comes from the ancient Greek ἀπόθεμα ("put away, put aside"), made of ἀπό ("off, away") and θέμα ("that which is laid down"), indicating a generic line written down.

The apothem a can be used to find the area of any regular n-sided polygon of side length s according to the following formula, which also states that the area is equal to the apothem multiplied by half the perimeter since ns = p. 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.

The following formulations are all equivalent: An apothem of a regular polygon will always be a radius of the inscribed circle.