Math
Regular Polygon Calculator
Calculate side length, apothem, circumradius, perimeter, area, and interior, exterior, and central angles of any regular polygon with 3 to 100 sides.
Regular polygon dimensions
Whole number from 3 to 100. Try 6 for a hexagon or 8 for an octagon.
Pick which length you know. The rest are derived.
Hexagon (6 sides)
259.8076
Area
- Side length (s)
- 10
- Perimeter
- 60
- Apothem (a)
- 8.6603
- Circumradius (R)
- 10
Interior angle
120°
Exterior angle
60°
Central angle
60°
Frequently Asked Questions about the Regular Polygon Calculator
What makes a polygon regular?
A polygon is regular when all of its sides are the same length and all of its interior angles are the same measure. The result is a shape with full rotational symmetry, so it fits perfectly inside one circle (the circumscribed circle) and exactly contains another (the inscribed circle). An equilateral triangle, a square, and a regular hexagon are the most familiar examples; a rectangle that is not a square is not regular because its sides are not all equal.
What is the difference between the apothem and the circumradius?
The apothem (a) is the perpendicular distance from the center to the midpoint of any side, so it is the radius of the inscribed circle. The circumradius (R) is the distance from the center to any vertex, the radius of the circumscribed circle. They are related by a = R cos(pi / n), so the apothem is always smaller than the circumradius and the ratio approaches 1 as the number of sides grows and the polygon nears a circle.
What is the area formula for a regular polygon?
The simplest form is area = 0.5 x apothem x perimeter, which works for any regular polygon once you know its apothem and side length. In closed form using only the side length s and the number of sides n: area = (1 / 4) x n x s^2 x cot(pi / n). For a hexagon with s = 10, that gives (1 / 4) x 6 x 100 x cot(30 degrees) = 150 x sqrt(3), about 259.81 square units.
Why do the interior angles always sum to (n - 2) x 180 degrees?
Any n-sided polygon can be split into (n - 2) triangles by drawing diagonals from a single vertex, and every triangle's angles sum to 180 degrees. That gives a total of (n - 2) x 180 degrees for the whole polygon. In a regular polygon all interior angles are equal, so each one measures (n - 2) x 180 / n degrees: 60 for a triangle, 90 for a square, 108 for a pentagon, 120 for a hexagon, and 135 for an octagon.
What are the names of regular polygons from 3 to 12 sides?
Equilateral triangle (3), square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), hendecagon or 11-gon (11), and dodecagon (12). Beyond that, mathematicians usually just say n-gon, so a 17-sided regular polygon is a regular 17-gon. The calculator labels 3 through 10 and 12 by name and falls back to the n-gon form for everything else.