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
What is the interior angle of a regular pentagon?
Each interior angle is exactly 108 degrees, and the sum of all interior angles is 540 degrees.