What is a regular polygon?

A regular polygon is a flat shape with equal side lengths and equal interior angles. Examples include the equilateral triangle (3 sides), square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), hendecagon (11), and dodecagon (12). All vertices lie on a common circle called the circumscribed circle.

How do you calculate the interior angle of a regular polygon?

The interior angle of a regular polygon with n sides is (n minus 2) times 180 degrees divided by n. For a hexagon (n = 6): (6 minus 2) times 180 divided by 6 = 4 times 180 divided by 6 = 120 degrees. The exterior angle is 360 divided by n.

What is the apothem of a regular polygon?

The apothem is the perpendicular distance from the centre of the polygon to the midpoint of any side. It equals half the side length divided by the tangent of pi divided by n, or equivalently s divided by (2 times tan(pi/n)). The apothem is used to calculate the area: area = 0.5 times perimeter times apothem.

Regular Polygon Calculator

A regular polygon is a flat, closed shape where every side has the same length and every interior angle is equal. Familiar examples include the equilateral triangle (3 sides), square (4 sides), pentagon (5 sides), hexagon (6 sides), and octagon (8 sides). Because all sides and angles are equal, you can calculate every property of the shape from just two pieces of information: the number of sides and the length of one side. The interior angle of a regular polygon is (n minus 2) times 180 degrees, divided by n, where n is the number of sides. The perimeter is simply n times the side length. The apothem, which is the perpendicular distance from the centre to the midpoint of any side, equals the side length divided by (2 times the tangent of pi over n). The area equals half the perimeter multiplied by the apothem, which can also be written as (n times s squared) divided by (4 times the tangent of pi over n). This calculator covers polygons from 3 sides (triangle) to 12 sides (dodecagon). You enter the number of sides and the side length, and the calculator returns the area, perimeter, interior angle, exterior angle, apothem, and circumradius. These values are used in engineering drawings, tile and paving design, architecture, and geometry coursework. Results are in the same unit as the side length you enter.

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Number of sides

Side length (s)

64.95

area (square units)

Perimeter30.00

Interior angle120.00°

Exterior angle60.00°

Apothem4.33

Circumradius5.00

How it works

For a regular polygon with n sides of length s: Interior angle = (n minus 2) times 180 divided by n degrees. Exterior angle = 360 divided by n degrees. Perimeter = n times s. Apothem = s divided by (2 times tan(pi/n)). Circumradius = s divided by (2 times sin(pi/n)). Area = 0.5 times perimeter times apothem, equivalently (n times s squared) divided by (4 times tan(pi/n)).

Worked example

A regular hexagon with side length 5. Interior angle = (6 minus 2) times 180 divided by 6 = 120.00 degrees. Perimeter = 6 times 5 = 30. Apothem = 5 divided by (2 times tan(30 degrees)) = 5 divided by 1.1547 = 4.33. Area = 0.5 times 30 times 4.33 = 64.95 square units. These match the default values above.

Related calculators

Hexagon Calculator: full property set for a regular hexagon including both diagonals.
Ellipse Calculator: area, circumference, and eccentricity of an ellipse.
Gear Ratio Calculator: gear ratios for polygonal gear profiles in mechanical systems.
Percent Error Calculator: check measurement accuracy for geometry coursework.

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