Apothem Calculator
Find the apothem — the distance from the centre of a regular polygon to the midpoint of a side — from the side length and number of sides.
How it works
Apothem = s ÷ (2·tan(π/n)) · Circumradius = s ÷ (2·sin(π/n))
The apothem is the perpendicular distance from the centre of a regular polygon to the midpoint of any side — the inradius of the polygon. Half of one side, the apothem and the circumradius form a right triangle with the half-central-angle π/n, so the side’s half length equals apothem·tan(π/n), which rearranges to the formula above. The apothem is always shorter than the circumradius (the centre-to-vertex distance).
Worked example
For a regular hexagon (n = 6) with side 8: apothem = 8 ÷ (2·tan(30°)) = 8 ÷ 1.1547 = 6.928 units, while the circumradius = 8 ÷ (2·sin(30°)) = 8 units.
Sources: Apothem — Wikipedia
Frequently asked questions
What is the difference between the apothem and the radius?
The apothem reaches from the centre to the middle of a side (the inradius), while the circumradius reaches from the centre to a vertex. The apothem is always the shorter of the two.
How is the apothem used to find area?
Area of a regular polygon equals ½ · perimeter · apothem. The apothem is the height of each of the triangular slices the polygon is divided into.
Does a square have an apothem?
Yes. For a square the apothem is half the side length, since tan(π/4) = 1, so apothem = s ÷ 2.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.