About the Polygon Calculator
A regular polygon is a shape whose sides are all the same length and whose angles are all equal. Pentagons, hexagons, octagons — and, less obviously, the square and the equilateral triangle, which are simply the regular polygons with four and three sides.
Everything about one follows from a single idea. Draw lines from the centre to each corner and the polygon divides into n identical isosceles triangles, all meeting at the middle. Solve one of those triangles and multiply by n, and you have the area, the perimeter, the height from the centre to a side, and the distance from the centre to a corner.
That is why every formula on this page carries a tangent or a sine of 180°/n: it is the same central triangle appearing again and again.
This calculator takes the number of sides plus any one length — the side, the apothem or the circumradius — and returns everything else.
How to Use the Polygon Calculator
Enter the number of sides first. Three or more, and a whole number.
Then choose what you know:
Side length. The usual case — one edge.
Apothem. The distance from the centre out to the middle of a side, at a right angle to it.
Circumradius. The distance from the centre to a corner.
Angles only. For when you want the angles and do not have any length. The calculator returns the angles and says plainly that the size is not determined.
The Formulas
Interior angle = (n - 2) × 180° / n
Exterior angle = 360° / n
Central angle = 360° / n
Perimeter P = n × s
Apothem a = s / ( 2 tan(180°/n) )
Circumradius R = s / ( 2 sin(180°/n) )
Area A = ½ × P × a
= n s² / ( 4 tan(180°/n) )
Diagonals n(n - 3) / 2The angle facts are worth knowing separately
The interior angles total (n − 2) × 180°. The reason is that any polygon cuts into n − 2 triangles, and each triangle contributes 180°. A quadrilateral is two triangles, so 360°. A pentagon is three, so 540°. For a regular polygon those are shared equally, so divide by n.
The exterior angles always total 360°, whatever n is. Walk once around the outside of any polygon and you finish facing the way you started, having turned through a full circle. Each corner turns you by one exterior angle, so they must add to 360. A triangle turns 120° at each corner, an octagon 45°, a hundred-sided polygon 3.6° — always totalling the same.
The central angle equals the exterior angle, which is not a coincidence: it is why the n triangles at the centre are identical.
Why area = ½ × perimeter × apothem
Each of the n central triangles has a side as its base and the apothem as its height, so its area is ½ × s × a. There are n of them, giving ½ × (n × s) × a = ½ × perimeter × apothem.
Compare that to a circle's area, πr², which can be written ½ × 2πr × r — ½ × circumference × radius. The same statement. The apothem is the polygon's radius, and as n grows the two formulas converge because the shapes do.
Reference Table
| Sides | Name | Interior | Exterior | Area for side = 1 | |-------|------|----------|----------|-------------------| | 3 | triangle | 60° | 120° | 0.4330 | | 4 | square | 90° | 90° | 1.0000 | | 5 | pentagon | 108° | 72° | 1.7205 | | 6 | hexagon | 120° | 60° | 2.5981 | | 8 | octagon | 135° | 45° | 4.8284 | | 10 | decagon | 144° | 36° | 7.6942 | | 12 | dodecagon | 150° | 30° | 11.196 |
Note how the area climbs steeply for the same side length. An octagon of side 1 holds nearly five times the area of a triangle of side 1, because more sides means more perimeter for a given side length.
The Hexagon Is Special
A regular hexagon's circumradius is exactly equal to its side length. Nothing else does this.
The reason is the central triangle again. With six sides the central angle is 60°, and the two equal sides make the base angles 60° too — so the triangle is equilateral, and all three lengths match. Six equilateral triangles, joined at a point.
This is why hexagons tile a plane with no gaps while pentagons cannot, and why honeycomb, graphene, basalt columns and steel mesh all end up hexagonal. Among the shapes that tile, the hexagon encloses the most area for the least perimeter — so a bee building in hexagons uses the least wax for the most honey.
Step-by-Step Example
A hexagonal patio, side 4 metres.
P = 6 × 4 = 24 m
a = 4 / (2 tan(30°)) = 4 / 1.1547 = 3.4641 m
A = ½ × 24 × 3.4641 = 41.569 m²Cross-check with the direct formula:
A = 6 × 4² / (4 tan(30°)) = 96 / 2.3094 = 41.569 m² ✓And the interior angle is (6 − 2) × 180 / 6 = 120°, which is the angle to cut each paving edge to.
An octagonal table from its circumradius. A table is to be 1.2 m from centre to corner, with eight sides.
s = 2 × 1.2 × sin(180°/8) = 2 × 1.2 × 0.38268 = 0.9184 m
a = 0.9184 / (2 tan(22.5°)) = 1.1086 m
A = ½ × (8 × 0.9184) × 1.1086 = 4.0729 m²Each edge is 918 mm, and the mitre at each joint follows from the 135° interior angle.
Understanding Your Result
The area is in square units of your input unit.
The perimeter shows the total with the side count and side length that made it, so you can check the recovered side length when you entered an apothem or radius.
The angles give all three. Interior is the angle inside each corner — the one you cut to. Exterior is the turn at each corner. Central is the angle at the middle between adjacent corners.
The apothem and circumradius are the two radii. The apothem is always the smaller of the two, because the middle of a side is closer to the centre than a corner is. They converge as n grows, and in the limit both become the radius of a circle.
The shape line names the polygon, counts its diagonals, and gives the percentage of its enclosing circle that it fills — 41% for a triangle, 83% for a hexagon, over 99.9% by a thousand sides. That last figure is Archimedes' method: he bounded π by squeezing the circle between inscribed and circumscribed polygons, working by hand up to 96 sides.
When Should You Use This Calculator?
Paving and decking. Hexagonal and octagonal patios need the area for material and the interior angle for the cuts.
Gazebos and summer houses. Usually hexagonal or octagonal; the side length and interior angle set out the frame.
Nuts and bolts. A hexagonal nut is specified across its flats, which is twice the apothem, and across its corners, which is twice the circumradius. Spanner sizes are the first of those.
Tiling patterns. Which polygons tile, and in what combination, comes down to whether their interior angles divide into 360°.
Craft and woodworking. Mitre angles for a polygonal frame are half the exterior angle at each joint.
Games and board design. Hex grids are built on the hexagon's equal-distance property.
Architecture. Polygonal towers, domes and bay windows all need this geometry.
Common Mistakes
Mixing up the apothem and the circumradius. The apothem goes to the middle of a side; the circumradius goes to a corner. They differ by a factor of cos(180°/n), and swapping them gives a wrong area in the right ballpark — the hardest kind of error to spot.
Using the interior angle where the exterior belongs. Mitre cuts and turns use the exterior angle. The interior angle is what sits inside the corner.
Assuming interior angles add to 360°. It is the exterior angles that always total 360. The interior ones total (n − 2) × 180, which grows with n.
Applying these formulas to an irregular polygon. Everything here assumes all sides and angles are equal. An irregular polygon needs its vertices individually — the shoelace formula on the coordinates, not a single side length.
Degrees against radians. The formulas use tan(180°/n) in degrees, equivalently tan(π/n) in radians. Mixing the two silently gives a wrong answer.
Expecting a polygon with fewer than three sides. Two straight sides cannot enclose anything.
Forgetting that a square is a regular polygon. If you enter 4 sides, this calculator gives exactly the square formulas — side squared for the area, half the side for the apothem. That is a useful way to sanity-check the general formulas against something you already know.