How do you calculate the area of a regular polygon?

The area of a regular polygon with n sides and side length a is A = (1/4) × n × a² × cot(π/n). Alternatively, if you know the inradius r, the area is A = n × r² × tan(π/n). Most polygon calculators compute this from either the side, inradius, or circumradius.

What is the interior angle of a regular polygon?

The interior angle of a regular polygon with n sides is ((n minus 2) × 180) ÷ n degrees. For a hexagon (6 sides) this is (4 × 180) ÷ 6 = 120 degrees. The sum of all interior angles equals (n minus 2) × 180 degrees.

What is the difference between inradius and circumradius?

The inradius (also called the apothem) is the radius of the largest circle that fits inside the polygon, touching each side at its midpoint. The circumradius is the radius of the smallest circle that contains the polygon, passing through each vertex. The circumradius is always larger than the inradius for any regular polygon.

How many sides does a polygon need to look like a circle?

A regular polygon begins to visually approximate a circle at around 30 to 40 sides. By 100 sides the difference is imperceptible to the naked eye. Mathematically, as the number of sides approaches infinity, the regular polygon approaches a perfect circle with area πr².

Regular Polygon Calculator

Regular Polygon Calculator | Area, Perimeter, Angles

Regular Polygon Calculator

Calculate all properties of any regular polygon. Choose the number of sides, enter any one measurement, and instantly find area, perimeter, inradius, circumradius, and angles.

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Regular Polygon Reference Table

Regular polygon properties by name, including interior angle and exterior angle for polygons from 3 to 12 sides.

Name	Sides (n)	Interior Angle	Exterior Angle	Sum of Angles
Trigon (Triangle)	3	60°	120°	180°
Tetragon (Square)	4	90°	90°	360°
Pentagon	5	108°	72°	540°
Hexagon	6	120°	60°	720°
Heptagon	7	128.57°	51.43°	900°
Octagon	8	135°	45°	1080°
Nonagon	9	140°	40°	1260°
Decagon	10	144°	36°	1440°
Undecagon	11	147.27°	32.73°	1620°
Dodecagon	12	150°	30°	1800°

Regular Polygon Formulas

Regular polygon formulas all derive from two key relationships: the number of sides (n) and one known measurement such as side length, inradius, or circumradius.

Side Length, Inradius, Circumradius

These three measurements are related by trigonometric functions. Given any one of them and the number of sides n, all others can be found using the formulas a = 2r·tan(π/n), r = R·cos(π/n), and R = a·csc(π/n)/2.

Area of a Regular Polygon

The area formula using side length is A = (1/4)·n·a²·cot(π/n). Using the inradius: A = n·r²·tan(π/n). Using the circumradius: A = (1/2)·n·R²·sin(2π/n). All three formulas produce the same result for a given polygon.

Interior and Exterior Angles

The interior angle of a regular n-gon equals ((n − 2) × 180°) ÷ n. The exterior angle equals 360° ÷ n. Interior and exterior angles always add up to exactly 180° at each vertex. As n increases, the interior angle approaches 180° and the exterior angle approaches 0°.

Incircle and Circumcircle Areas

The incircle has area π·r² where r is the inradius. The circumcircle has area π·R² where R is the circumradius. The ratio of these areas is always (cos(π/n))², which is fixed for any given polygon type.

Common Regular Polygons in the Real World

Regular polygons appear throughout nature, architecture, engineering, and everyday objects.

The equilateral triangle (3 sides) appears in truss bridges, triangular tiles, and musical instruments. The square (4 sides) is the most common shape in architecture, floor plans, and screens. The pentagon (5 sides) is found in the US Department of Defense headquarters and in certain flower petals. The hexagon (6 sides) occurs naturally in honeybee combs, basalt columns, and snowflake structures. The octagon (8 sides) is used for stop signs and the Octagonal Hall in Rome. Regular polygon tessellations, where polygons tile a plane with no gaps, are only possible with triangles, squares, and hexagons.

Frequently Asked Questions

A regular polygon is a closed shape where all sides are equal in length and all interior angles are equal in measure. Examples include equilateral triangles (3 sides), squares (4 sides), pentagons (5 sides), and hexagons (6 sides).

The area of a regular polygon with n sides and side length a is A = (1/4) × n × a² × cot(π/n). If you know the inradius r, use A = n × r² × tan(π/n). Most polygon calculators compute area from the side, inradius, or circumradius.

The interior angle of a regular n-sided polygon is ((n − 2) × 180) ÷ n degrees. For a hexagon this is (4 × 180) ÷ 6 = 120 degrees. The sum of all interior angles equals (n − 2) × 180 degrees.

The inradius (apothem) is the radius of the largest circle fitting inside the polygon, touching each side at its midpoint. The circumradius is the radius of the circle passing through each vertex. The circumradius is always larger than the inradius.

A regular polygon begins to visually approximate a circle at around 30 to 40 sides. By 100 sides the difference is imperceptible. Mathematically, as the number of sides approaches infinity, the polygon approaches a perfect circle with area πr².