How to Find the Area of a Regular Pentagon

The formula

A = 5s² ÷ (4 × tan(36°))

The tangent makes this look more forbidding than the hexagon formula, but it collapses to a single constant: 4 × tan(36°) is about 2.9062, so the whole thing is roughly 1.7205 × s².

For a pentagon with sides of 6: A ≈ 1.7205 × 36 ≈ 61.94 square units.

Why there is a tangent in it

Like the hexagon, a regular pentagon splits into triangles from its centre — five of them this time, each with a 36° half-angle at the top.

The difference is that these triangles are not equilateral. Five lots of 72° fill the 360° around the centre, but 72° is not 60°, so the triangles lean. To find their height you need trigonometry, and that is where tan(36°) enters.

This is the general pattern for regular polygons. The hexagon is the lucky case where the angle works out to 60° and the trigonometry disappears. Every other regular polygon needs a tangent somewhere.

The general formula for any regular polygon

If you would rather learn one formula than five, here it is. For a regular polygon with n sides of length s:

A = n × s² ÷ (4 × tan(180° ÷ n))

Put n = 5 and you get the pentagon formula. Put n = 6 and it simplifies down to the hexagon one. Put n = 4 and it collapses all the way to s², the area of a square, because tan(45°) = 1.

It is worth checking that last one yourself once. Watching a general formula reduce to something you already know is the quickest way to trust it.

Perimeter

Five equal sides, so P = 5 × s. For our example, 5 × 6 = 30.

Using the apothem instead

If a question gives you the apothem — the distance from the centre to the middle of a side — there is a neater route.

A = (perimeter × apothem) ÷ 2

This works for any regular polygon, not just pentagons, and it is much easier than the tangent version. It is the same idea as a triangle's "half base times height", applied to all n triangles at once.

If you are given the apothem, use this. If you are given only the side, use the formula at the top of the page.