Pentagon Perimeter Calculator
A regular pentagon has five equal sides and five equal angles, so its perimeter is five times the side length. Enter a side length and this calculator returns the perimeter, the area, the apothem (the distance from the center to a side midpoint) and the diagonal. The diagonal of a regular pentagon equals the side times the golden ratio, one of the most famous relationships in geometry. Every formula here is an exact result that needs no measured constant; just keep your length units consistent.
Pentagon formulas
Perimeter = 5 x side
Area = (1/4) x square root of (5 x (5 + 2 x square root of 5)) x side squared
Apothem = side / (2 x tangent of 36 degrees)
Diagonal = side x golden ratio (approximately 1.618)
The area constant is approximately 1.720 and the apothem factor about 0.688. The diagonal-to-side ratio is exactly the golden ratio, (1 + square root of 5) / 2.
Pentagon geometry context
- A regular pentagon has five equal sides and interior angles of 108 degrees.
- The diagonal equals the side times the golden ratio, about 1.618.
- Area also equals one half times perimeter times apothem.
- Unlike hexagons, regular pentagons cannot tile a flat plane alone.
- Use one length unit for the side; area is reported in that unit squared.
Pentagon perimeter: frequently asked questions
What is the perimeter of a regular pentagon?
A regular pentagon has five equal sides, so its perimeter is five times the side length: P = 5s. For a side of 6 units the perimeter is 5 x 6 = 30 units. This calculator also returns the area, the apothem and the diagonal for the same side.
How do you find the area of a regular pentagon?
The area of a regular pentagon is one quarter of the square root of 5 times (5 plus 2 times the square root of 5), all multiplied by the side squared. That constant is approximately 1.720, so the area is about 1.720 times s squared. It is an exact value derived from the pentagon's geometry.
What is the apothem of a pentagon?
The apothem is the perpendicular distance from the center to the midpoint of a side. For a regular pentagon it equals the side divided by twice the tangent of 36 degrees, which is about 0.688 times the side. The apothem is useful because area also equals one half times perimeter times apothem.
How long is the diagonal of a regular pentagon?
The diagonal of a regular pentagon equals the side multiplied by the golden ratio, approximately 1.618. This elegant relationship is why pentagons and the golden ratio appear together so often in geometry and design.
Does this work for an irregular pentagon?
No. The formulas assume a regular pentagon with five equal sides and equal angles. For an irregular pentagon, add the five side lengths for the perimeter, and use a method such as splitting it into triangles to find the area.
Official sources
- U.S. National Institute of Standards and Technology: Digital Library of Mathematical Functions.
- NASA: Geometry and area reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.