Calculate area and perimeter of a regular pentagon.
The Pentagon Area Calculator is a geometric tool designed to calculate the area, perimeter, and diagonal lengths of a regular pentagon. A pentagon is a five-sided polygon. A regular pentagon has five equal sides (s), five equal interior angles of 108 degrees, and is closely associated with the golden ratio (φ ≈ 1.618) through its diagonals.
The area (A) of a regular pentagon can be calculated using the side length (s) with the formula: A = 1/4 × √(5 × (5 + 2√5)) × s² ≈ 1.720477 × s². Alternatively, it can be computed using the apothem (a), which is the perpendicular distance from the center to the midpoint of a side: A = 5/2 × s × a. The perimeter (P) of the pentagon is calculated by multiplying the side length by five: P = 5s. The diagonal length (d) between any two non-adjacent vertices is d = s × φ = s × (1 + √5) / 2.
Regular pentagons appear in architecture (such as the Pentagon building in the United States), design (patterns on soccer balls), and natural structures (the cross-section of certain flowers and fruits). The pentagon area calculator automates these complex radical equations, providing instant, decimal-precise results that help designers, engineers, and students solve geometric layouts easily.
Properties of a Regular Pentagon
A regular pentagon is a five-sided polygon with all equal sides and equal internal angles of 108°.
- Area (A):
A = 1/4 × √(5 × (5 + 2√5)) × s² ≈ 1.720477 × s² - Perimeter:
P = 5 × s
How it Works & Formula
Computes the geometric area and boundary perimeter of a regular five-sided pentagon.
Practical Examples
Pentagonal layout with side 100m: Area ≈ 17204.8 m².
Frequently Asked Questions
Each interior angle is exactly 108 degrees, and the sum of all interior angles is 540 degrees.
The sum of interior angles of an n-sided polygon is (n - 2) × 180°. For a pentagon (n=5): (5 - 2) × 180° = 540°.
A regular pentagon has all five sides equal and all five angles equal (108° each). An irregular pentagon has sides and/or angles of varying lengths or measures.
Famous examples include the U.S. Pentagon building in Washington D.C. and the shapes of home plates in baseball fields.
The apothem is the distance from the center to the midpoint of any side. For a regular pentagon with side s, the apothem = s / (2 × tan(36°)) ≈ 0.6882s.