Regular Polygon Math, From Interior Angles to Area and Apothem

A regular polygon is the simplest kind of many-sided shape: every side the same length, every corner the same angle. Triangles, squares, pentagons, hexagons, all the way up to a near-circular hundred-sided figure. Despite how plain they look, regular polygons hide a small web of measurements that depend on each other, and getting one wrong throws the rest off. This is a practical walk through those numbers, what each one means, and where they show up off the page.

If you just want the answers, the Regular Polygon Calculator takes the number of sides and one side length and returns the whole set at once. But knowing where the figures come from is what lets you sanity-check them, and that is the part worksheets and CAD tutorials tend to skip.

Interior and exterior angles

Start with the corners. Every interior angle of a regular polygon with n sides equals:

interior angle = (n − 2) · 180 / n

The reason is tidy. Any polygon splits into n − 2 triangles by drawing diagonals from one corner, and each triangle holds 180 degrees, so the angles total (n − 2) · 180. Spread that evenly across n identical corners and you get the formula above. A triangle gives 60, a square 90, a pentagon 108, and a hexagon 120. As n climbs, the interior angle creeps toward 180 and the shape rounds out into something that looks like a circle.

The exterior angle is the turn you make at each corner when you walk once around the outline:

exterior angle = 360 / n

Interior and exterior always add to 180 at any single corner. And there is a fact worth keeping in your pocket: the exterior angles of any convex polygon sum to exactly 360 degrees, no matter how many sides. One full lap around the shape turns you through one complete circle. That is why the exterior angle is often the easier of the two to reason about.

Area from one side length

The angles are free once you know n. To get area you need a size, and the cleanest input is a single side length s. The formula is:

area = (1/4) · n · s² · cot(π / n)

The cotangent term carries all the shape information, encoding how widely the corners open as n grows. One thing trips people up constantly: that π / n is in radians, not 180 / n in degrees. Set a calculator to the wrong mode and the area comes out nonsense. If you are nesting this inside other geometry work, a scientific calculator with an explicit radian or degree toggle saves a lot of silent errors.

A quick check that the formula behaves: a square is n = 4, s = 2, giving (1/4) · 4 · 4 · cot(45°) = 4 · 1 = 4, which is exactly s², as a square should be.

A worked hexagon

Let me run a full example, because seeing the numbers land together is what makes the relationships stick. Take a regular hexagon with side length s = 10.

Interior angle: (6 − 2) · 180 / 6 = 4 · 180 / 6 = 120 degrees.
Exterior angle: 360 / 6 = 60 degrees. Add it to the interior angle and you get 180, as expected.
Sum of interior angles: (6 − 2) · 180 = 720 degrees, shared as 720 / 6 = 120 per corner.
Area: (1/4) · 6 · 10² · cot(π / 6) = 1.5 · 100 · cot(30°) = 150 · 1.7320508 ≈ 259.8 square units.

So a hexagon with 10-unit sides covers about 259.8 square units. The first time I needed this for real was sizing a hexagonal paver pattern for a patio, and I distinctly remember second-guessing whether the area should scale with s or s² (it is s², since both the base and height of every internal triangle grow with s). Plugging two side lengths into the tool and watching the area quadruple when I doubled s settled it faster than re-deriving anything.

Apothem and circumradius

Two more measurements close out the set, and they are the ones people mix up most. The apothem is the distance from the center to the midpoint of a side. It is also called the inradius, since it is the radius of the inscribed circle that just touches every side:

apothem = s / (2 · tan(π / n))

The circumradius reaches from the center out to a vertex, the radius of the circle that passes through every corner:

circumradius = s / (2 · sin(π / n))

The circumradius is always the larger of the two, because a corner pokes farther from the center than the midpoint of an edge does. For our hexagon with s = 10, the apothem is about 8.66 and the circumradius is exactly 10 (a hexagon is the special case where the circumradius equals the side length). Confuse these two and you will undersize whatever you are building, so it is worth picturing: apothem to the flat of a side, circumradius to the point.

Where this shows up off the page

These figures are not just homework fodder. A few places they earn their keep:

Tiling and flooring. Only triangles, squares, and hexagons tile a flat plane with no gaps, and the reason is the interior angle: 60, 90, and 120 all divide evenly into 360, so copies meet cleanly at a vertex. A pentagon's 108 does not, which is why you never see a plain pentagonal tile floor.
Furniture and garden builds. Laying out a hexagonal tabletop or an octagonal planter, the apothem tells you how far the center sits from each edge and the circumradius tells you how far the corners reach, the two numbers you need to mark on the ground before cutting.
Nuts, bolts, and flanges. A hex nut is a regular hexagon, and the wrench size measures across the flats, which is twice the apothem. Drilling a six-hole bolt circle on a flange, the circumradius is your bolt-circle radius and the central angle between holes is the exterior angle, 360 / 6 = 60 degrees.
Logos and design grids. Snapping a logo onto a clean polygonal frame means matching vertices to a circumradius and edges to an apothem, the same two radii in a different costume.

When a build mixes inches and millimetres, run the side length through a unit converter first so every output lands in one system, since the area scales with the square of whatever unit you feed in.

Pulling it together

The whole shape follows from two inputs: the number of sides n and one side length s. From n alone come both angles and the angle sum; add s and you unlock perimeter, area, apothem, and circumradius. Keep π / n in radians inside the trig, keep the apothem and circumradius straight, and the numbers will always reconcile. When you want to skip the arithmetic and see the figure drawn to scale, the Regular Polygon Calculator does every step in your browser and hands you a shareable link to the exact shape.

Made by Toolora · Updated 2026-06-13