Area of a regular polygon: formula, calculator and worked example

What the symbols mean

$A$: the area
$n$: the number of sides (3 or more)
$s$: the length of each side
$tan (π/n)$: the tangent of $π/n$ radians (that is $180°/n$ )

When to use it

Use it for hexagonal tiles and nuts, octagonal stop signs and gazebos, pentagons, and any regular shape where you know a side. If you know the apothem (the distance from the centre to the middle of a side), the half-perimeter-times-apothem form is quicker.

Worked example

A hexagonal tile has 6 sides, each 2 cm long. What is its area? (n = 6, s = 2)

Start from the formula.
$A = \frac{n s^{2}}{4 \tan (\frac{π}{n})}$
Substitute $n$ = $6$ , $s$ = $2$ and work it out.
$A = \frac{6 \cdot 2^{2}}{4 \tan (\frac{π}{6})}$
$= \frac{6 \cdot 4}{4 \tan (\frac{π}{6})}$
$= \frac{24}{4 \tan (\frac{π}{6})}$
$\approx 10.3923$
(The angle π/n is in radians: $\frac{π}{6}$ radians is 30°.)

Answer A ≈ 10.39 cm²

Why it works

Join the centre to every corner: the polygon splits into n identical isosceles triangles, each with base s. The height of each triangle is the apothem, $s/(2tan (π/n))$ , from the right triangle with angle $π/n$ at the centre. Adding the n triangle areas gives the formula.

Common mistakes

Using degrees in tan when your calculator is in radian mode, or the other way round: $π/n$ radians is $180°/n$ .
Applying it to irregular polygons. Only equal sides and equal angles give n identical triangles.

Area of a regular polygon

Calculator

See it move

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes