Pentagon Perimeter Calculator (regular)

P = 5 × s

Five equal sides. A regular pentagon with 6 cm sides has perimeter 30 cm.

Where pentagon perimeters show up in measurements:

• Pentagonal gazebos and bandstands. A 6-ft-side pentagonal gazebo has 30 ft of perimeter, which sizes the decking trim, the rail and the base framing all at once.
• Home plate in baseball. Five sides, and the back is a POINT rather than an edge: 17 in across the front, 8.5 in down each side, then two 12 in edges meeting at that point. 17 + 8.5 + 8.5 + 12 + 12 = 58 in. Note it is not a regular pentagon, so P = 5s does not apply.
• The Pentagon building. Each outer wall is 921 ft, so the outer perimeter is 4,605 ft, or 0.87 miles. The outer pentagon encloses 1,459,000 sq ft, about 33 acres, though the building itself is five nested rings around a courtyard rather than a solid slab.
• Pentagonal cupolas, rooftop lanterns and bandstand roofs, where five equal rafters meet at a peak.
• Pentagonal planting beds and patios, where the perimeter sets the edging order.

Worked example: decking a pentagonal gazebo

A 5-ft side pentagonal gazebo. Perimeter = 5 × 5 = 25 ft of trim around the deck. Floor area = 1.7205 × 25 = 43 sq ft.

For the railing, count five corner posts plus one mid-post per side, so ten uprights in all. Each rail runs the full 5 ft between corners and every joint is cut at 36°, half the 72° turn you make at each corner.

Worked example: walking round the Pentagon

The outside of the Pentagon, in Arlington, Virginia, measures 4,605 ft, which is 1.40 km or 0.87 miles. At a steady 3 mph that is a 17-minute lap. The number people usually quote instead is the corridor walk inside, where the five nested rings and the radial spokes mean no two offices are more than about seven minutes apart on foot.

Other measurements from the side s (for a regular pentagon):

• Perimeter: P = 5s
• Apothem (inradius, centre to the midpoint of a 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 (where φ is the golden ratio)
• Area: A ≈ 1.7205 × s²
• Interior angle: 108°

The golden ratio φ = (1 + √5) / 2 shows up because the diagonals of a regular pentagon divide each other in the golden ratio. This is the geometric foundation for the pentagram (five-pointed star) that fits inside a regular pentagon.

Quick sanity check. The circumscribed circle has radius 0.8507s, so its diameter is 1.7013s. Divide the perimeter by that diameter and you get 5 / 1.7013 = 2.94.

For a circle the same ratio is π, 3.14. A hexagon gives 3.00, an octagon 3.06, and a hundred-sided polygon 3.1411. That climb toward π is exactly how Archimedes computed it, by squeezing a circle between inscribed and circumscribed polygons and adding sides until the two closed in on each other.