Pentagon Area Calculator (regular)

A regular pentagon has five equal sides and five equal angles (108° each). Given just the side length s, every other measurement follows.

Area formula:

A = (1/4) × √(25 + 10√5) × s² ≈ 1.7205 × s²

A 10 cm regular pentagon has area 172.05 cm². A 1-inch pentagon has area 1.7205 sq in.

Equivalent compact form:

A = (5/4) × s² × cot(π/5)

The cot(π/5) factor is about 1.376, and the 5/4 gives the final 1.7205. Both forms work; the first is easier to compute without a calculator that has cotangent.

Where pentagons appear in real measurements:

• Home plate in baseball is a pentagon. Specifically, a 17-inch-wide pentagon shape: 17 in wide at the front, with the back corner pointed toward the catcher. The area is roughly 211 sq in.
• The Pentagon building in Arlington, Virginia. Each outer wall is 921 feet long. Total floor area: about 6.5 million sq ft (the building has multiple stories — the single-floor pentagonal footprint is about 1.3 million sq ft).
• Soccer ball panels. A traditional soccer ball uses 12 black pentagons and 20 white hexagons in a truncated-icosahedron pattern.
• Some country flags feature pentagon shapes — usually 5-pointed stars rather than filled pentagons.
• Flower petals. Many flowers (rose, hibiscus, geranium) have five-fold symmetry approximating a regular pentagon.

Worked example — gazebo floor:

A small pentagonal gazebo with 6 ft sides. Area = 1.7205 × 36 = 61.94 sq ft of floor. That fits a small table and three chairs comfortably.

Other useful measurements from the same side s:

• Apothem (inradius — center to mid-side): r = s / (2 × tan(36°)) ≈ 0.6882 × s
• Circumradius (center to vertex): R = s / (2 × sin(36°)) ≈ 0.8507 × s
• Diagonal: d = s × φ ≈ 1.618 × s — yes, the golden ratio appears in regular pentagon geometry, and that’s no coincidence.

Why the golden ratio shows up: the ratio of any diagonal to a side in a regular pentagon is exactly the golden ratio φ = (1 + √5) / 2. The pentagon’s diagonals form a 5-pointed star (a pentagram) with golden-ratio relationships throughout. Pythagoras’s followers studied this for that reason.

Sanity-check the area: the pentagon fits inside its circumscribed circle of radius R ≈ 0.851s. That circle has area π × (0.851s)² ≈ 2.27 × s². Our pentagon area 1.72s² uses about 76% of the circle area — which makes sense geometrically.