Math Geometry • Basic Shapes and Properties
Compute apothem (inradius), circumradius, central angle, perimeter, and area for any regular polygon.
Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(),
tan(), ln(), log(), abs(). Use * for multiplication.
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The apothem r is the distance from the center of a regular polygon to the midpoint of any side. It is also the radius of the inscribed circle (inradius).
For a regular n-gon with side length s, the circumradius is R = s / (2 sin(pi/n)). This comes from a right triangle formed by the center, a vertex, and the midpoint of a side.
They are linked by r = R cos(pi/n). The factor cos(pi/n) depends only on the number of sides n.
The central angle is the angle at the polygon’s center between two adjacent vertices. It equals 360/n degrees (or 2 pi/n radians).
A common formula is A = (1/2) x P x r, where P is the perimeter and r is the apothem. Since P = n x s, the same relationship can be used after solving for the missing quantities.