Pentagon Calculator

A regular pentagon has five equal sides and five equal interior angles of 108°. Given the side length a, every other measurement follows from it directly.

Perimeter = 5a

Diagonal = a × φ = a × (1 + √5) / 2 ≈ 1.618a

The golden ratio φ appears because the diagonals of a regular pentagon divide each other in the golden ratio — this is not a coincidence. It is why pentagons show up in art and architecture tied to ideas about proportion.

Apothem (inradius — center to midpoint of a side) = a / (2 tan(π/5)) ≈ 0.6882a Circumradius (center to a corner) = a / (2 sin(π/5)) ≈ 0.8507a

Area = (5a² / 4) × cot(π/5) = (5a²) / (4 tan(36°)) ≈ 1.7205a²

Or equivalently: Area = (1/4) × √(25 + 10√5) × a²

At a = 10 cm, the area is 172.05 cm², the diagonal is 16.18 cm, and the apothem is 6.88 cm.

Where pentagons appear in real measurements: the home plate in baseball is a pentagon with specific dimensions (17 in wide), the Pentagon building in Virginia has 5 sides each 921 feet long (area ≈ 603,000 sq ft of floor space — the campus footprint is much larger), and many flowers have five-fold symmetry that approximates the regular pentagon.

The interior angle sum is always 540° for any pentagon, regular or not. The 108° per interior angle for the regular case follows from (5−2) × 180° / 5 = 108°.