Regular Polygon Calculator, Explained

Give one measurement of a regular polygon and read off every other one: side, perimeter, area, apothem, circumradius, both angles, and how many diagonals it has.

What a regular polygon is

A regular polygon has n equal sides and n equal interior angles. A square is the regular 4-gon. A stop sign is a regular octagon (n = 8). Because every side and angle is identical, a single number fixes the whole shape. Tell the calculator the side length of a hexagon and it can report the area, the two radii and the angles without any further input.

Two circles frame every regular polygon. The circumscribed circle passes through all the vertices; its radius is the circumradius R. The inscribed circle touches the middle of every side; its radius is the apothem a. The apothem is always smaller than the circumradius. For a regular hexagon with side 4, you will see R = 4 and a \approx 3.464 below.

When to use it, and when not

Use this calculator whenever the shape is truly regular: floor tiles, bolt heads, gazebo footprints, gears drawn as regular polygons, dartboard segments. One measurement is enough, so it saves you from chaining trig by hand.

Do not use it when the sides or angles differ. A rectangle that is not a square, a general pentagon from surveyed corners, or any irregular lot needs coordinates instead. For those, reach for the Polygon Area Calculator, which works from vertex points. If you only have a triangle, the Triangle Solver handles all cases including scalene ones.

An equilateral triangle is regular, but a "right triangle" or an isosceles triangle is not. If your three sides are not all equal, this tool will give wrong numbers. Send unequal triangles to the triangle tools instead.

The formulas and where they come from

Split a regular n-gon into n identical isosceles triangles by drawing lines from the center to each vertex. Each triangle has apex angle 2\pi / n at the center. That single triangle drives every formula.

a = \frac{s}{2} \cot\left(\frac{\pi}{n}\right)

Here s is the side length and a is the apothem. The apothem is the height of one of those triangles, so it drops from the center to the midpoint of a side.

R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)}

The circumradius R reaches from the center to a vertex. It is the slanted side of the same triangle.

A = \frac{1}{2} n s a = \frac{1}{4} n s^{2} \cot\left(\frac{\pi}{n}\right)

The area A is just n copies of one triangle, each with base s and height a. The compact form A = \frac{1}{2} P a, with perimeter P = n s, is worth remembering: area equals half the perimeter times the apothem.

The angles depend only on n, not on size. The interior angle at each vertex is

\theta_{\text{int}} = \frac{(n - 2) \cdot 180^{\circ}}{n}

and the exterior angle (the turn you make walking around the outline) is 360^{\circ} / n. The two add to 180^{\circ}. The number of diagonals is

D = \frac{n(n - 3)}{2}

which counts every vertex pair minus the n sides.

Worked example: a hexagon with side 4

Reproducing the demo

Set n = 6, known quantity Side length, and value 4, then press Calculate.

Perimeter: P = 6 \times 4 = 24.
Apothem: a = \frac{4}{2}\cot(30^{\circ}) = 2 \times 1.7321 = 3.4641.
Circumradius: R = \frac{4}{2\sin(30^{\circ})} = \frac{4}{2 \times 0.5} = 4. In a regular hexagon the circumradius equals the side, a handy check.
Area: A = \frac{1}{2} \times 24 \times 3.4641 = 41.57. Cross-check with \frac{1}{4} \times 6 \times 16 \times \cot(30^{\circ}) = 24 \times 1.7321 = 41.57.
Interior angle: \frac{(6-2) \times 180}{6} = 120^{\circ}. Exterior angle: 360 / 6 = 60^{\circ}.
Diagonals: \frac{6 \times 3}{2} = 9.

So the demo hexagon has perimeter 24, apothem 3.464, circumradius 4, area 41.57, interior angle 120°, and 9 diagonals.

How area grows with the number of sides

Fix the circumradius at R = 1 and increase n. The polygon fills more of its circle. Its area is A = \frac{1}{2} n R^{2} \sin(2\pi / n), which climbs toward the circle's area of \pi \approx 3.1416.

Area rises fast at first (the triangle covers only 1.299 of the circle's 3.1416) then flattens as the polygon hugs the circle.

Notice the jump from 1.299 at n = 3 to 2.598 at n = 6, then only 3.106 at n = 24. Each extra side adds less than the last. By n = 24 the polygon already fills 98.9% of the circle. This is exactly why cutting a circle into many straight segments looks smooth.

Reading and interpreting the results

The five size measurements are all proportional to each other. Double the side and the perimeter, apothem and circumradius all double, while the area quadruples because it scales with the square of length. So a hexagon with side 8 has area 41.57 \times 4 = 166.3, not 83.14.

Apothem: Center-to-side distance. Use it as the "radius" when you need the largest circle that fits inside the polygon.
Circumradius: Center-to-vertex distance. Use it to find the smallest circle that contains the polygon, or the bolt-circle diameter for evenly spaced holes.
Interior angle: The angle inside each corner. Depends only on n. A regular pentagon is always 108°.
Exterior angle: The turn at each corner, 360^{\circ}/n. Useful for miter cuts: two boards meeting at a corner are each cut at half of it.

For a practical link between the two radii, their ratio is fixed by n: a / R = \cos(\pi / n). For the hexagon that is \cos(30^{\circ}) = 0.8660, so 3.4641 / 4 = 0.8660 matches.

Common mistakes

The errors below account for most wrong answers.

Mixing radius and apothem. If a spec says a hexagon "fits in a circle of diameter 8", that is the circumradius 4, not the apothem. Choosing the wrong one shifts the area by a factor of \cos^2(\pi/n), here 25%.
Degrees versus radians. The formulas use \pi / n in radians. If you compute by hand, keep your calculator in the right mode. \cot(30) in radian mode is nonsense.
Forgetting the square law for area. When you scale from a solved example, remember area goes with length squared.
Using it on an irregular shape. Covered above, but it is the single most common misuse.

For miter joints, take the exterior angle and halve it. A hexagonal frame has exterior angle 60°, so each of the two mating boards is cut at 30°. An octagon gives 45° / 2 = 22.5°.

Related calculators

If your shape is not regular, or you want a different view of the same geometry, these tools help:

Polygon Area Calculator for any polygon from its corner coordinates.
Circle Calculator for the inscribed and circumscribed circles themselves.
Triangle Solver and the Right Triangle Calculator for single triangles, including the isosceles wedge inside your polygon.
Trigonometry Calculator to evaluate the \cot and \sin terms in the formulas.
3D Solid Geometry Calculator when a polygon becomes the base of a prism or pyramid.
Coordinate Geometry Toolkit for distances and midpoints if you place your polygon on a grid.

Frequently asked questions

What is the difference between the apothem and the radius?

The apothem reaches the middle of a side; the circumradius (often just called the radius) reaches a vertex. The apothem is always the smaller of the two. Their ratio is \cos(\pi/n), so for a hexagon the apothem is 86.6% of the circumradius.

Can I start from the area instead of the side?

Yes. Pick Area as the known quantity. The calculator inverts the area formula to get the side, s = \sqrt{\frac{4A \tan(\pi/n)}{n}}, then computes everything else. Any one of the five measurements works as the starting point.

Why does a regular hexagon have circumradius equal to its side?

Because its central triangles are equilateral. The apex angle is 360^{\circ}/6 = 60^{\circ}, and with two equal sides of length R the base equals R too. That base is the polygon side, so s = R. It is unique to the hexagon.

What does the polygon become as n gets very large?

It approaches its circumscribed circle. At n = 100 the area is 3.1395 against the circle's 3.1416, a gap of 0.07%. This is the classical way of approximating \pi.

How many diagonals does a regular polygon have?

Use D = n(n-3)/2. A pentagon has 5, a hexagon 9, an octagon 20. The count depends only on n, not on size.

Try it yourself in the tool