Area of a Regular Polygon Formula

Geometry Formulas: Area, Volume, Perimeter
Formula:

What is the formula for the area of a regular polygon?

Answer

Area of a regular polygon
A = (n · a²) / (4 · tan(π / n))

Here n is the number of sides, a is the side (or base). The result comes out in square units.

Where the formula comes from

A = (n · a²) / (4 · tan(π / n))

Split the regular n-gon into n identical triangles from the centre. Each has base a and height a / (2·tan(π/n)) — the apothem — so the whole is n·a²/(4·tan(π/n)). For n = 6 this collapses to the hexagon formula, which is the check that it is right.

Worked examples

Let n = 5, a = 6. Then the area of a regular polygon ≈ 61.94 square units.

Let n = 8, a = 3.5. Then the area of a regular polygon ≈ 59.15 square units.

The common mistake

Applying it to an irregular polygon. This formula assumes every side and every angle is equal; an irregular polygon has to be cut into triangles and added up instead.

Other quantities: Regular polygon

Related Calculations

See Also

Geometry Formulas: Area, Volume, Perimeter

These are the formulas for the area, perimeter, circumference, volume and surface area of the standard shapes — 31 of them. Each page shows not only the formula but where it comes from, what every letter stands for, two worked examples and the mistake that this particular shape invites. For example, it can help you find out what is the formula for the area of a regular polygon? (The answer is: A = (n · a²) / (4 · tan(π / n))). Pick the pair — "Area of a circle", for instance — and hit the "Find out" button.
Geometry Formulas: Area, Volume, Perimeter
Formula:

Values from the Formula (a = 6)

If n =Area
3≈ 15.59 square units
436 square units
5≈ 61.94 square units
6≈ 93.53 square units
8≈ 173.82 square units
10≈ 276.99 square units
12≈ 403.06 square units

FAQ

What is the formula for the area of a regular polygon?

Area of a regular polygon A = (n · a²) / (4 · tan(π / n))

Where does this formula come from?

Split the regular n-gon into n identical triangles from the centre. Each has base a and height a / (2·tan(π/n)) — the apothem — so the whole is n·a²/(4·tan(π/n)). For n = 6 this collapses to the hexagon formula, which is the check that it is right.

What is the most common mistake here?

Applying it to an irregular polygon. This formula assumes every side and every angle is equal; an irregular polygon has to be cut into triangles and added up instead.

How do I use this formula?

Put your own numbers in place of the letters. With n = 8, a = 3.5, for instance, the formula A = (n · a²) / (4 · tan(π / n)) gives ≈ 59.15 square units.

See Also