About the Regular Polygon Calculator
A regular polygon has all its sides the same length and all its angles equal: the equilateral triangle, the square, the regular pentagon, hexagon, octagon and so on. Once you know how many sides it has and one measurement, every other property follows. This calculator takes the number of sides and any one of the side length, circumradius (centre to corner), apothem (centre to the middle of a side), perimeter or area, and gives you all the rest, including the interior and exterior angles, the sum of the angles and the number of diagonals. It works for anything from 3 to 1,000 sides in any unit.
Updated October 2026
How to use the Regular Polygon Calculator
- 1Enter the number of sides, for example 6 for a hexagon.
- 2Choose which measurement you know: side, circumradius, apothem, area or perimeter.
- 3Type its value in any unit.
- 4Read the area and perimeter, then the table of other measurements.
- 5Copy the summary line.
Common polygons at a glance
- Triangle (3): interior angle 60°.
- Square (4): 90°.
- Pentagon (5): 108°.
- Hexagon (6): 120°.
- Octagon (8): 135°.
- Decagon (10): 144°.
A worked example
A designer wants a regular octagonal table top that fits inside a circle of radius 50 cm. Choose 8 sides, select circumradius and enter 50. The side length is 2 × 50 × sin(22.5°), about 38.27 cm, the apothem is about 46.19 cm, the perimeter is 306.1 cm and the area is about 7,071 square centimetres. The corners at 135 degrees can be cut accurately with a mitre saw set to 22.5 degrees.
A useful check is to start from the circumradius for polygons that must fit inside a circle, such as a table top cut from a round board or a nut that fits a round hole, and from the apothem for polygons that must fit around a circle, such as a gazebo built around a round floor. The apothem is always smaller than the circumradius, and the ratio between them approaches 1 as the number of sides grows, which is why a polygon with many sides looks nearly round.
If you are drawing or cutting a polygon by hand, the exterior angle tells you how far to turn at each corner: 360 divided by the number of sides. A hexagon turns 60 degrees at every corner, an octagon 45. The central angle, the angle at the centre between two neighbouring corners, is the same figure, which gives a simple way to mark equal divisions around a circle with a protractor, for example when laying out bolt holes or garden stakes.
Where regular polygons appear
Nuts and bolts are hexagons, stop signs are octagons, honeycomb cells are hexagons and many gazebos, tiles and logos are built from regular shapes. As the number of sides increases the polygon gets closer to a circle, which is how early mathematicians estimated pi by comparing the perimeters of polygons with 96 sides.
Related tools
The Circle Calculator, Triangle Calculator and Area Calculator cover other flat shapes, the Pyramid Calculator and Cone Calculator extend to solids, and the Trigonometry Calculator handles the angle sums behind the formulas.
Features
- Any regular polygon from 3 to 1,000 sides
- Start from side, radius, apothem, area or perimeter
- Interior and exterior angles
- Sum of interior angles and number of diagonals
- Named shapes up to the dodecagon
- Works in any unit
Frequently asked questions
What is the area formula for a regular polygon?
Area equals half the perimeter times the apothem. Equivalently, with n sides of length s, it is n times s squared divided by 4 times the tangent of 180 degrees over n. A hexagon with 10 unit sides has an area of about 259.8.
What is the apothem?
The apothem is the shortest distance from the centre to a side, meeting it at a right angle. It is the radius of the circle that fits inside the polygon touching every side.
What is the circumradius?
It is the distance from the centre to a corner, the radius of the circle that passes through all the corners. For a hexagon it equals the side length.
How do I find the interior angle of a regular polygon?
Take the number of sides minus 2, multiply by 180 and divide by the number of sides. A regular pentagon has (5 − 2) × 180 ÷ 5 = 108 degrees at each corner.
How many diagonals does a polygon have?
n × (n − 3) ÷ 2. A hexagon has 9, an octagon 20 and a decagon 35.
Can I find the side from the area?
Yes. Choose area as the known measurement and the calculator works back to the side length, then fills in everything else.