Regular Polygon Calculator
Enter the number of sides and one measurement: the side length, apothem, circumradius or area. You get the area, perimeter, interior and exterior angles and more.
Worked examples
A hexagon
6 sides, side 10
Area 259.81, perimeter 60, apothem 8.66
Interior angle 120 degrees.
A pentagon
5 sides, side 1
Area 1.7205
Interior angle 108 degrees, 5 diagonals.
About Regular Polygon Calculator
A regular polygon has all its sides equal and all its angles equal, like a square, a regular hexagon or a stop sign octagon. The apothem is the distance from the centre to the middle of a side, and the circumradius is the distance from the centre to a corner. For n sides of length s, the apothem is s divided by 2 tan(π/n), and the area is half the apothem times the perimeter.
Each interior angle is (n - 2) × 180 divided by n degrees, so a hexagon has 120 degrees at every corner, and the exterior angles always add up to 360 degrees. The number of diagonals is n(n - 3) divided by 2. Enter the number of sides and any one of the side length, apothem, circumradius or area, and the page finds the others. A hexagon with sides of 10 has an area of about 259.8 and a perimeter of 60, which helps when you are cutting tiles, laying paving or designing a nut or gazebo.
How to use Regular Polygon Calculator
- Enter the number of sides, from 3 upwards.
- Choose what you know: the side length, apothem, circumradius or area.
- Enter that measurement.
- Read the area, perimeter and the table of lengths and angles.
Common uses
- Cutting hexagonal or octagonal tiles and boards
- Designing a gazebo, bay window or planter
- Geometry homework on polygons
- Finding the size of a bolt head or nut across the flats
Frequently asked questions
What is the apothem?
It is the shortest distance from the centre of the polygon to a side, meeting the side at a right angle. It is the height of each of the triangles the polygon can be split into.
How many degrees are in each angle of a regular polygon?
Each interior angle is (n - 2) × 180 divided by n. A pentagon has 108 degrees, a hexagon 120 and an octagon 135.
How many diagonals does a polygon have?
n × (n - 3) divided by 2. A hexagon has 9 and an octagon has 20.
How do I find the side from the area?
The page does this when you choose area as the known value. It uses the area formula n × s² divided by (4 tan(π/n)), solved for s.
Is a square a regular polygon?
Yes, it is the regular polygon with 4 sides. A rhombus that is not a square is not, because its angles are not all equal.