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Calculator Answers

Regular Polygon Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A regular polygon calculator is a mathematical utility that computes the geometric properties of equilateral and equiangular polygons. By entering the number of sides and either the side length, circumradius, inradius (apothem), or area, the calculator determines all remaining dimensions. It calculates the perimeter, area, sum of interior angles, and individual exterior angles. Students and graphic designers use this tool to construct symmetric shapes, analyze geometric relationships, and solve trigonometry problems.

Enter the number of sides and the side length. The calculator returns the perimeter, area, apothem, and interior and exterior angles of the regular polygon.

Quick Answer

Calculate properties of any regular polygon. Enter the number of sides and one known dimension to find the area, perimeter, and angles.

Whole number 3 or greater. · e.g. 6

Positive number. · e.g. 5

A regular polygon has all sides equal and all interior angles equal. Triangle is n=3, square is n=4, hexagon is n=6.

Hexagon

Area

64.9519

n = 6, s = 5

Sides (n)6
Side length (s)5
Perimeter30
Apothem4.3301
Area64.951905
Interior angle120°
Exterior angle60°

Perimeter = n × s · Area = (n × s²) ÷ (4 × tan(π ÷ n)) · Apothem = s ÷ (2 × tan(π ÷ n)) · Interior angle = (n − 2) × 180 ÷ n.

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Examples

Hexagon, side 5

perimeter 30 · area ≈ 64.952 · apothem ≈ 4.330 · interior 120°

Pentagon, side 4

perimeter 20 · area ≈ 27.528 · apothem ≈ 2.753 · interior 108°

Octagon, side 3

perimeter 24 · area ≈ 43.456 · apothem ≈ 3.621 · interior 135°

How it works

A regular polygon has every side and every angle equal. From n (number of sides) and s (side length) every other measurement follows by a single tangent-based formula.

Perimeter · P = n × s

Area · A = (n × s²) ÷ (4 × tan(π ÷ n))

Apothem · a = s ÷ (2 × tan(π ÷ n))

Interior angle = (n − 2) × 180 ÷ n · Exterior angle = 360 ÷ n

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Frequently asked questions

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.