Regular Polygon Calculator

A regular polygon is a closed shape with all sides equal in length and all interior angles equal. An equilateral triangle (3 sides), square (4 sides), pentagon (5), hexagon (6), heptagon (7), and octagon (8) are common examples.

The apothem is the perpendicular distance from the center of a regular polygon to the middle of any side. It is shorter than the radius (which goes to a vertex) and lies inside the polygon. The apothem is used in the area formula A = ½ × perimeter × apothem.

For a regular polygon with n sides, each interior angle equals ((n − 2) × 180°) ÷ n. A triangle's interior angle is 60°, a square's is 90°, a hexagon's is 120°, and so on. The sum of interior angles is (n − 2) × 180°.

Each exterior angle is 360° ÷ n. The exterior angles of any polygon (regular or not) sum to 360°. For a regular hexagon, each exterior angle is 60°.

A = (n × s²) ÷ (4 × tan(π ÷ n)), where n is the number of sides and s is the side length. The same formula can be written A = ½ × perimeter × apothem, which is geometrically clear: the polygon decomposes into n congruent triangles each with area ½ × s × apothem.

No, a circle is the limit of a regular polygon as n approaches infinity, but a single n value cannot represent it. For circle math (circumference, area), use the circle calculator. For very large n the polygon's perimeter approaches the circle's circumference and the area approaches the circle's area.