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Polygon Calculator — Regular Polygon Properties

Calculate area, perimeter, angles, and radii for any regular polygon

Area

259.81

Perimeter

60.00

Shape

Hexagon

Regular Hexagon

259.81

Area (sq units)

Perimeter

60.00

units

Diagonals

9

Sides

6

Angles

Interior Angle120.00°
Exterior Angle60.00°
Sum of Interior Angles720°

Radii

Apothem (inradius)8.6603
Circumradius10.0000

Polygon Shape

Area

259.81

Perimeter

60.00

Apothem

8.6603

Circumradius

10.0000

Formulas Used

Polygon Area

Area = n × s² / (4 × tan(π/n))

The area enclosed by a regular polygon with n sides of length s.

Where:

n= Number of sides
s= Length of each side
π= Pi, approximately 3.14159

Interior Angle

Interior Angle = (n − 2) × 180° / n

Each interior angle of a regular polygon. The sum of all interior angles is (n−2) × 180°.

Where:

n= Number of sides

Apothem

Apothem = s / (2 × tan(π/n))

The distance from the center of the polygon to the midpoint of any side (also called the inradius).

Where:

s= Length of each side
n= Number of sides

Circumradius

Circumradius = s / (2 × sin(π/n))

The distance from the center of the polygon to any vertex.

Where:

s= Length of each side
n= Number of sides

Example Calculations

1Regular Hexagon with Side 10

Inputs

Number of Sides6
Side Length10

Result

Area259.81
Perimeter60.00
Interior Angle120.00°
Apothem8.6603
Circumradius10.0000
Diagonals9

Area = 6 × 100 / (4 × tan(π/6)) = 600 / 2.3094 = 259.81. Interior angle = (6−2) × 180/6 = 120°. A regular hexagon with side 10 has a circumradius equal to its side length.

2Regular Pentagon with Side 8

Inputs

Number of Sides5
Side Length8

Result

Area110.11
Perimeter40.00
Interior Angle108.00°
Apothem5.5055
Circumradius6.8052
Diagonals5

Area = 5 × 64 / (4 × tan(π/5)) = 320 / 2.9062 = 110.11. Interior angle = (5−2) × 180/5 = 108°. A pentagon has exactly 5 diagonals, equal to its side count.

3Regular Octagon with Side 5

Inputs

Number of Sides8
Side Length5

Result

Area120.71
Perimeter40.00
Interior Angle135.00°
Apothem6.0355
Circumradius6.5328
Diagonals20

Area = 8 × 25 / (4 × tan(π/8)) = 200 / 1.6569 = 120.71. Interior angle = (8−2) × 180/8 = 135°. Stop signs are regular octagons.

Frequently Asked Questions

Q

How do you calculate the area of a regular polygon?

The area of a regular polygon with n sides of length s is calculated using Area = n × s² / (4 × tan(π/n)). This formula works for any regular polygon from triangles to 100-gons. For a regular hexagon with side 10, the area is 6 × 100 / (4 × tan(30°)) = 259.81 square units.

  • Triangle (3 sides, s=10): area = 43.30 sq units
  • Square (4 sides, s=10): area = 100.00 sq units
  • Pentagon (5 sides, s=10): area = 172.05 sq units
  • Hexagon (6 sides, s=10): area = 259.81 sq units
  • Octagon (8 sides, s=10): area = 482.84 sq units
PolygonSidesArea (s=10)Interior Angle
Triangle343.3060°
Square4100.0090°
Hexagon6259.81120°
Octagon8482.84135°
Dodecagon121,119.62150°
Q

What is the interior angle of a regular polygon?

Each interior angle of a regular polygon = (n−2) × 180° / n, where n is the number of sides. The exterior angle is simply 360° / n. Interior and exterior angles always sum to 180°. As side count increases, interior angles approach 180°.

  • Triangle: (3−2) × 180/3 = 60° per angle, 180° total
  • Square: (4−2) × 180/4 = 90° per angle, 360° total
  • Hexagon: (6−2) × 180/6 = 120° per angle, 720° total
  • Decagon: (10−2) × 180/10 = 144° per angle, 1,440° total
  • A circle can be thought of as a polygon with infinite sides and 180° interior angles
PolygonInterior AngleExterior AngleAngle Sum
Triangle60°120°180°
Square90°90°360°
Pentagon108°72°540°
Hexagon120°60°720°
Octagon135°45°1,080°
Q

What is the apothem and circumradius of a polygon?

The apothem is the distance from the center to the midpoint of a side (inradius). The circumradius is the distance from the center to a vertex. For a regular polygon with side s: apothem = s / (2 × tan(π/n)) and circumradius = s / (2 × sin(π/n)).

  • Hexagon (s=10): apothem = 8.6603, circumradius = 10.0000
  • Square (s=10): apothem = 5.0000, circumradius = 7.0711
  • The apothem equals the circumradius only for a circle (infinite sides)
  • Area can also be calculated as: Area = ½ × perimeter × apothem
  • As side count increases, apothem and circumradius converge
Q

How many diagonals does a regular polygon have?

The number of diagonals in any polygon = n(n−3)/2, where n is the number of sides. A triangle has 0 diagonals, a square has 2, a pentagon has 5, and a hexagon has 9. The formula grows quadratically with the number of sides.

  • Triangle (3 sides): 3(0)/2 = 0 diagonals
  • Square (4 sides): 4(1)/2 = 2 diagonals
  • Pentagon (5 sides): 5(2)/2 = 5 diagonals
  • Hexagon (6 sides): 6(3)/2 = 9 diagonals
  • Decagon (10 sides): 10(7)/2 = 35 diagonals
PolygonSidesDiagonalsTotal Line Segments
Triangle303
Square426
Pentagon5510
Hexagon6915
Decagon103545
Q

What is the difference between regular and irregular polygons?

A regular polygon has all sides equal in length and all interior angles equal. An irregular polygon has sides or angles of different sizes. This calculator handles regular polygons only. Examples of regular polygons include equilateral triangles, squares, and regular hexagons.

  • Regular: all sides equal + all angles equal (e.g., equilateral triangle, square)
  • Irregular: sides or angles differ (e.g., rectangle, scalene triangle)
  • Only regular polygons have a defined apothem and circumradius from center
  • Regular polygons are always convex; irregular polygons can be concave
  • Regular polygon area formulas require only side count and side length

Understanding Regular Polygon Properties

A regular polygon is a closed shape where all sides have equal length and all interior angles are equal. From equilateral triangles to dodecagons, regular polygons appear throughout architecture, nature, and engineering.

The key properties of any regular polygon can be derived from just two values: the number of sides (n) and the side length (s). The interior angle formula (n−2) × 180°/n reveals how angles increase as sides increase, approaching 180° for very large n.

The apothem (center to side midpoint) and circumradius (center to vertex) define the inscribed and circumscribed circles. The area formula n × s² / (4 × tan(π/n)) is equivalent to ½ × perimeter × apothem, connecting all measurements.

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Last Updated: Mar 9, 2026

This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.

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