Math · Geometry
Regular polygon calculator.
Hexagon, octagon, pentagon or any n-gon: side, apothem, circumradius, area and angles from one measurement.
Inputs
Polygon
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How do you find the area of a regular polygon?
A regular polygon has n equal sides and equal angles. With side length a, the apothem (centre to the middle of a side) is r = a / (2 tan(π/n)) and the circumradius (centre to a corner) is R = a / (2 sin(π/n)). The area is half the perimeter times the apothem, A = ½ × n a × r. Each interior angle is (n − 2) × 180° / n, and there are n(n − 3) / 2 diagonals.
Worked example. A regular hexagon with a side of 10 has an apothem of 8.6603, a circumradius of 10 (equal to the side, a hexagon property), a perimeter of 60, an area of 259.8076, interior angles of 120° and 9 diagonals. A regular pentagon with circumradius 10 has a side of 11.7557 and an area of 237.7641.
Frequently asked questions
What is the area formula for a regular polygon?
A = ½ × perimeter × apothem, or A = ¼ n a² / tan(π/n) from the side length alone.
What is the apothem?
The distance from the centre to the midpoint of a side, which is also the radius of the inscribed circle. The circumradius is the distance to a corner.
How many degrees in each angle of a regular polygon?
Interior angle = (n − 2) × 180 ÷ n. A hexagon has 120°, a pentagon 108°, an octagon 135°.
Can I enter the area and get the side?
Yes. Choose Area under “I know the” and the calculator works back to the side, apothem and circumradius.
Sources
Checked
Side, inradius (apothem), circumradius, area and interior angle formulas are from the MathWorld Regular Polygon page. The diagonal count n(n-3)/2 is the standard combinatorial count (choose any two of n corners, minus the n sides). Runs entirely in your browser. Nothing you type is sent or stored.
Scope
Results are exact mathematics rounded for display, so they are only as good as the measurements you enter. Any length unit works as long as every value uses the same one; areas come out in that unit squared and volumes in that unit cubed. For homework, DIY layout and estimating, not a substitute for a surveyor, engineer or your teacher's required method.