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, but not a regular one, so the formula above does not apply to it. It is 17 in across the front, 8.5 in down each side, then two 12 in edges meeting at a point aimed at the catcher. Area = 17 × 8.5 for the front block plus ½ × 12 × 12 for the point, which is 216.5 sq in.
• The Pentagon building in Arlington, Virginia. Each outer wall is 921 feet, so the outer pentagon covers 1.7205 × 921² = 1,459,000 sq ft, about 33 acres. Its floor area is far larger, roughly 6.5 million sq ft, because the building is five storeys of concentric rings around an open courtyard rather than a solid pentagon.
• Soccer ball panels. The traditional ball uses 12 black pentagons and 20 white hexagons in a truncated-icosahedron pattern. The pentagons are the reason it curves: hexagons alone would stay flat forever.
• Wild roses, hibiscus and geranium. Five-fold symmetry is everywhere in flowering plants, and okra in cross-section is a passable regular pentagon.
• Pentagonal gazebos and cupolas, where the 108° corner sets a 36° mitre on every piece of trim.

Worked example: a 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, centre to mid-side): r = s / (2 × tan(36°)) ≈ 0.6882 × s
• Circumradius (centre to vertex): R = s / (2 × sin(36°)) ≈ 0.8507 × s
• Diagonal: d = s × φ ≈ 1.618 × s, the golden ratio, and no coincidence.

Why the golden ratio shows up: the ratio of any diagonal to a side in a regular pentagon is exactly φ = (1 + √5) / 2. Draw all five diagonals and they trace a pentagram enclosing a smaller upside-down pentagon, sides 1/φ² of the original, which does the same thing again. The Pythagoreans studied this shape hard for exactly 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 ≈ 2.275 × s². The pentagon’s 1.7205s² fills about 76% of it, against 83% for a hexagon and 64% for a square. Each extra side pushes the polygon closer to the circle.