Pentagon Area Calculator
A regular pentagon is a five-sided polygon with all sides of equal length and all interior angles of 108 degrees. This calculator computes the area, perimeter, circumradius, and apothem of a regular pentagon from the side length. The area formula involves the golden ratio and equals approximately 1.72048 times the square of the side length. The perimeter is simply five times the side length. The circumradius is the radius of a circle that passes through all five vertices, and the apothem is the radius of a circle that touches the midpoint of all five sides. Regular pentagons appear throughout nature and are important in geometry and mathematics.
Pentagon formulas
Area = (s^2 * sqrt(25 + 10*sqrt(5))) / 4
Area (approximate) = 1.72048 * s^2
Perimeter = 5 * s
Circumradius (R) = s * sqrt(50 + 10*sqrt(5)) / 10
Circumradius (approximate) = 0.85065 * s
Apothem (a) = s * sqrt(25 + 10*sqrt(5)) / 10
Apothem (approximate) = 0.68819 * s
Example calculation
For a regular pentagon with side length 10:
- Area = 1.72048 * 10^2 = 172.05 square units
- Perimeter = 5 * 10 = 50 units
- Circumradius = 0.85065 * 10 = 8.51 units
- Apothem = 0.68819 * 10 = 6.88 units
Pentagon area calculator: frequently asked questions
What is a regular pentagon?
A regular pentagon is a five-sided polygon with all sides of equal length and all interior angles equal to 108 degrees. It is one of the most common regular polygons and appears frequently in nature, art, and architecture.
What is the formula for the area of a regular pentagon?
The area of a regular pentagon with side length s is: Area = (s^2 * sqrt(25 + 10*sqrt(5))) / 4, which simplifies to approximately 1.72048 * s^2. This formula comes from dividing the pentagon into five triangles from the center.
What is the interior angle of a regular pentagon?
Each interior angle of a regular pentagon is 108 degrees. The sum of all five interior angles is 540 degrees, which follows the formula (n-2)*180 for an n-sided polygon.
What are the circumradius and apothem of a pentagon?
The circumradius (radius of the circumscribed circle) is approximately 0.85065 * s, where s is the side length. The apothem (radius of the inscribed circle) is approximately 0.68819 * s.
Where do regular pentagons appear in nature?
Regular pentagons appear in starfish, sea urchins, some flowers, and the internal structure of apples. The golden ratio appears in pentagon geometry, making it mathematically and aesthetically significant.
Official sources
- Wolfram MathWorld: Regular Pentagon.
- ISO 80000-2: Mathematical signs and symbols.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.