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 turns up because the diagonals of a regular pentagon cut each other in exactly that ratio. Draw all five and they enclose a smaller upside-down pentagon, which does the same thing again, and again. That self-similarity is why the pentagon and the pentagram keep appearing wherever people have been thinking about proportion.

Apothem (inradius, centre to the midpoint of a side) = a / (2 tan(π/5)) ≈ 0.6882a Circumradius (centre 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.

The one thing a pentagon cannot do

It will not tile a plane. Squeeze three regular pentagons around a point and you get 3 × 108° = 324°, leaving a 36° gap; four would need 432° and overlap. That failure is exactly why a football needs 12 pentagons scattered among its 20 hexagons: the hexagons tile flat forever, and the pentagons are the defects that force the sheet to curve into a ball.

Worked example: a pentagonal gazebo floor

Five 8 ft sides. Perimeter = 40 ft of rail. Area = 1.7205 × 64 = 110.1 sq ft, roughly a 10 by 11 ft room. The apothem is 5.51 ft, so the floor joists span 11.01 ft flat to flat, and the corner-to-centre circumradius is 6.81 ft. Every corner is a 108° interior angle, which means a 36° mitre on the trim.

Pentagons in the wild

Baseball’s home plate is a pentagon, though not a regular one: 17 in across the front, two 8.5 in sides, and two 12 in sides meeting at a point. The formulas above do not apply to it.

The Pentagon building in Virginia does have five equal outer walls, each 921 ft long. That makes its outer footprint 1.7205 × 921² ≈ 1,459,000 sq ft, about 33 acres. Its floor space is far larger still, roughly 6.5 million sq ft, because the building is five storeys of concentric rings around a central courtyard rather than a solid pentagon.

Five-fold symmetry is common in flowers and starfish, and it is the standard shape of an okra pod in cross-section.

The interior angle sum is 540° for any pentagon, regular or not. The regular case gives 108° each, from (5 − 2) × 180° / 5.