Pentagon Area Calculator

Formula

A = (√(25 + 10√5) / 4) × l² ≈ 1.720 × l²

Area of a regular pentagon

A regular pentagon carries five equal sides and five interior angles that each measure 108°. From the side length L, you can find the area with A = (5/4)·L²·cot(π/5) ≈ 1.7204·L². The same number comes out of A = (5/2)·L·a, where a is the apothem, the perpendicular distance from the center out to each side. Try L = 10 m and the area lands at 1.7204 · 100 ≈ 172.05 m².

There's a deep link between the regular pentagon and the golden ratio. Divide a diagonal by a side and you land right on φ ≈ 1.618. Euclid worked out a construction using only straightedge and compass in Book IV of the Elements, which puts the pentagon among the handful of regular polygons you can actually draw with classical tools.

Applications and variants

The shape lends its name to the Pentagon, the US Department of Defense headquarters. It's also the building block of the classic black-and-white soccer ball (12 black pentagons and 20 white hexagons, a truncated icosahedron), and you'll meet it again in countless flowers, in starfish, and in the five-pointed star, or pentagram, you can inscribe inside it. When all you have is the side, the apothem comes from a = L / (2·tan(π/5)) ≈ 0.6882·L.

FAQ

What if the pentagon is irregular? The 1.7204·L² shortcut is for regular pentagons only. For an irregular one, break it into triangles or run the shoelace formula over the coordinates of its vertices.

How do I get the perimeter? All five sides are the same length, so just multiply: P = 5·L.

What is the apothem for L = 10? Work it out as a = 10 / (2·tan(36°)) ≈ 6.88 m. Drop that back into (5/2)·L·a and you recover the same ~172 m².

Is the pentagon related to the golden ratio? Very much so. The diagonal-to-side ratio is φ, and so is the ratio in which two diagonals split each other where they cross inside the pentagon.