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 — used for sizing decking, rail trim, and base framing.
• Home plate in baseball. A standardized pentagon: 17 in wide at the front, 8.5 in on each angled corner, 12 in on each side, 17 in on the back. Total perimeter ≈ 58 in.
• The Pentagon building. Each outer wall is 921 ft, total perimeter = 4,605 ft = 0.87 mi. The walking perimeter inside is roughly the same (the floor plan is 1.3 million sq ft per floor).
• Pentagonal architectural cupolas, observatories, and rooftop features.
• Some country flags use 5-pointed stars, which involve pentagon geometry.

Worked example — pentagonal gazebo decking:

A 5-ft side pentagonal gazebo. Perimeter = 5 × 5 = 25 ft. That’s the trim length around the deck — but the floor area is 1.7205 × 25 = 43 sq ft.

For a railing around the gazebo, every other side might have a railing post or balustrade. Five sides means five posts at corners plus mid-posts on each side — about 10 to 15 vertical supports total.

Worked example — Pentagon building perimeter loop:

Walking around the outside of the Pentagon (in Washington, DC) is 4,605 ft, or about 1.5 km. A leisurely walking pace of 3 mph covers this in 17 minutes. Hardly anyone walks it casually; the security perimeter is more like a 1-mile rectangular loop around the property.

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

• Perimeter: P = 5s
• Apothem (inradius — center to midpoint of 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 perimeter (5s) is always greater than the diameter of the circumscribed circle (≈ 1.7 × s × 2 = 3.4s). Specifically the ratio of perimeter to diameter of circumscribed circle is 5 / (2 × 0.851) ≈ 2.94 — close to π (≈ 3.14) because a pentagon is “trying” to be a circle. The more sides, the closer to π.