Regular Polygon Calculator

How it works

A regular polygon has n equal sides and n equal corners, so it can be cut into n identical isosceles triangles meeting at the centre. Each triangle has a corner angle of 360°/n at the centre; splitting it down the middle gives a right triangle whose legs are half a side and the apothem, and whose hypotenuse is the circumradius. Every other measurement follows from that one right triangle with its half-angle π/n (that is, 180°/n).

From the side, the apothem is half the side divided by tan(180°/n) and the circumradius is half the side divided by sin(180°/n). Given the apothem or circumradius instead, the same relationships run backwards to give the side first. The area is n times the area of one centre triangle, which works out to half the perimeter times the apothem.

The angle at each corner is (n − 2) × 180° / n because the interior angles of any n-sided polygon total (n − 2) × 180°. The exterior angle, the turn you make walking round each corner, is 360°/n, and the two always add to 180°. Each corner connects to n − 3 others by a diagonal, and each diagonal has two ends, so there are n(n − 3)/2 of them.

Formula

a = s ÷ (2·tan(π/n))        R = s ÷ (2·sin(π/n))
s = 2·a·tan(π/n)            s = 2·R·sin(π/n)
perimeter = n·s
area      = n·s² ÷ (4·tan(π/n)) = ½ · perimeter · a
interior angle = (n − 2)·180° ÷ n     exterior angle = 360° ÷ n
diagonals = n·(n − 3) ÷ 2

Example

A regular hexagon with 10 ft sides: tan(30°) = 0.5774, so the apothem is 10 ÷ (2 × 0.5774) = 8.6603 ft, and sin(30°) = 0.5, so the circumradius is 10 ÷ 1 = 10 ft (in a hexagon the circumradius always equals the side). The perimeter is 60 ft and the area is 6 × 100 ÷ (4 × 0.5774) = 259.8076 sq ft, the same as ½ × 60 × 8.6603. Each interior angle is 120°, each exterior angle 60°, and there are 6 × 3 ÷ 2 = 9 diagonals.

A regular pentagon with 10 ft sides has an apothem of 10 ÷ (2 × tan 36°) = 6.8819 ft, a circumradius of 10 ÷ (2 × sin 36°) = 8.5065 ft and an area of 172.0477 sq ft; its interior angles are 108°.

Assumptions and limitations

• The polygon is regular: every side the same length and every corner the same angle. For an irregular shape use the polygon-from-coordinates or irregular quadrilateral calculators.
• All lengths are in the unit you choose and the area is in that unit squared. The unit labels the results; nothing is converted.
• Angles are in degrees.
• Results are rounded for display only.

Frequently asked questions