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Regular Pentagon Area Calculator

Find the area of a regular pentagon.

Result

61.94 units²
30.00 units

How it is calculated

For a regular pentagon with five equal sides and equal angles, the area from the side length s is A = ¼ · √(5 · (5 + 2√5)) · s², which is about 1.72048 × s². The perimeter is simply P = 5 · s. The side must be greater than zero. The tool prints both values with two decimals and the unit "units", so the area is in the square of whatever length unit you typed (cm gives cm², m gives m²). It works only for regular pentagons; an irregular five-sided shape needs to be split into triangles.

Worked example

A regular pentagon has side s = 6. The constant is ¼ · √(5 · (5 + 2√5)) ≈ 1.72048. Square the side: 6² = 36. Area: 1.72048 × 36 = 61.94 units². Perimeter: 5 × 6 = 30.00 units. The area is 61.94 square units.

Try this example →

Common mistakes

  • Applying the formula to an irregular pentagon. It needs five equal sides and equal angles; unequal sides give a wrong area.
  • Entering the distance from the centre to a corner (circumradius) or to a side's midpoint (apothem) as the side length. Only the side itself is accepted.
  • Forgetting the area is in square units. A side in metres gives m², and mixing it with a cm-based length gives results off by 10,000.
  • Doubling the side and expecting double the area. Area scales with the square, so it becomes four times larger.

About this calculator

This calculator finds the area and perimeter of a regular pentagon (five equal sides and angles) from its side length s: Area = ¼·√(5(5+2√5))·s², a formula derived from splitting the pentagon into five congruent isosceles triangles from its center and using the apothem (the distance from center to the midpoint of a side). Perimeter is simply 5·s.

Regular pentagons and their defining ratio are closely tied to the golden ratio φ ≈ 1.618 — the ratio of a pentagon's diagonal to its side is exactly φ — which is why pentagonal geometry appears in art, architecture, and design (like the Pentagon building, home plate in baseball being a modified pentagon, and pentagonal floor tiles).

Geometry students computing regular polygon areas, architects and designers working with pentagonal layouts, and hobbyists building pentagon-based crafts or models all use this calculator, entering just the side length.

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