About the Regular Polygon Calculator
This regular polygon calculator solves any shape with equal sides and equal angles — equilateral triangles, squares, pentagons, hexagons, octagons and beyond, up to 1,000 sides. Enter the number of sides and whichever single measurement you know (side length, circumradius, apothem, area or perimeter) and it returns every other property.
You get the side length, perimeter, area, apothem (the distance from the centre to the middle of a side), circumradius (centre to a corner), the interior, exterior and central angles, the sum of interior angles and the number of diagonals. That covers geometry homework as well as practical jobs like cutting an octagonal table top, laying out a hexagonal gazebo or sizing a hex nut.
All lengths use whatever unit you enter, and area is in the matching square unit. Angles are in degrees. The calculator assumes a convex regular polygon; star polygons are not covered.
With the default inputs, the area is 259.8076. Change any value above to recalculate instantly.
How to use the regular polygon calculator
- 1Enter the number of sides — 6 for a hexagon, 8 for an octagon and so on.
- 2Choose which measurement you already know.
- 3Enter that value in any unit.
- 4Read the area, side, perimeter, apothem, radius and angles.
Formula and method
A regular polygon can be split into n identical isosceles triangles meeting at the centre, each with a base equal to the side s and a height equal to the apothem a. Each triangle’s apex (central) angle is 360°/n, so half of it is π/n radians, which gives a = s ÷ (2·tan(π/n)) and the circumradius R = s ÷ (2·sin(π/n)). The area is n triangles of ½·s·a, i.e. ½ × perimeter × apothem.
Whatever measurement you enter, the calculator first converts it to the side length using these relationships, then derives everything else. Interior angles sum to (n − 2)·180°, each exterior angle is 360°/n, and the number of diagonals is n(n − 3)/2.
- n
- Number of sides
- s
- Side length
- a
- Apothem (centre to the midpoint of a side)
- R
- Circumradius (centre to a vertex)
- P
- Perimeter = n·s
Worked examples
Hexagon with 10 cm sides
A regular hexagon’s circumradius equals its side, so R = 10. The apothem is 10 ÷ (2·tan 30°) ≈ 8.66, and the area is ½ × 60 × 8.66 ≈ 259.81 cm². Each interior angle is 120°.
Octagon table 24 in across the flats
Across the flats is twice the apothem, so a = 12. Side = 2 × 12 × tan 22.5° ≈ 9.94 in, and the top covers about 477.2 sq in. Across the corners is 2 × 12.99 ≈ 25.98 in.
Pentagon with an area of 100
Solving A = 5s² ÷ (4·tan 36°) for s gives s = √(4 × 100 × tan 36° ÷ 5) ≈ 7.62, so the perimeter is about 38.12.
Dodecagon inscribed in a circle of radius 10
Side = 2 × 10 × sin 15° ≈ 5.176. A regular 12-gon inscribed in a circle of radius R has area exactly 3R², here 300.
Frequently asked questions
How do you find the interior angle of a regular polygon?+
Use (n − 2) × 180° ÷ n. A hexagon has (6 − 2) × 180 ÷ 6 = 120°, an octagon 135°, and a pentagon 108°.
What is an apothem?+
The apothem is the perpendicular distance from the centre of a regular polygon to the midpoint of any side. It is also the radius of the largest circle that fits inside the polygon.
What is the area of a regular hexagon?+
A = (3√3 ÷ 2) × s², about 2.598 × s². A hexagon with 10 cm sides therefore has an area of roughly 259.8 cm².
How many diagonals does a polygon have?+
n(n − 3) ÷ 2. A pentagon has 5, a hexagon 9 and an octagon 20. Each vertex connects to every other vertex except itself and its two neighbours.
What angle do I cut for an octagon frame?+
Cut each end at half the exterior angle: 360° ÷ 8 ÷ 2 = 22.5°. In general the mitre angle is 180° ÷ n for a regular n-sided frame.