Regular Pentagon Calculator

Calculate area, perimeter, apothem, diagonal, and interior angles of a regular pentagon.

Enter Side Length

s

Result

Area
--
square units
Perimeter--
Apothem--
Diagonal--
Interior Angle--
Circumradius--

Step-by-Step Solution

A = (s^2 * sqrt(25 + 10*sqrt(5))) / 4

Understanding the Regular Pentagon

A regular pentagon is a five-sided polygon where all sides are equal in length and all interior angles are equal (108 degrees each). Pentagons appear extensively in nature, architecture, and design, from the famous Pentagon building to the faces of a dodecahedron.

Regular Pentagon Formulas

Area

The area of a regular pentagon with side length s:

A = (s^2 * sqrt(25+10*sqrt(5))) / 4

Perimeter

Sum of all five equal sides:

P = 5s

Apothem

Distance from center to the midpoint of a side:

a = s / (2 * tan(pi/5))

Diagonal

The diagonal of a regular pentagon relates to the golden ratio:

d = s * (1 + sqrt(5)) / 2

The Golden Ratio Connection

The regular pentagon has a deep connection to the golden ratio (phi = (1 + sqrt(5))/2, approximately 1.618). The ratio of the diagonal to the side length of a regular pentagon is exactly the golden ratio. This mathematical relationship makes the pentagon aesthetically pleasing and gives it unique geometric properties.

Interior and Exterior Angles

  • Interior angle: Each interior angle of a regular pentagon is 108 degrees.
  • Sum of interior angles: (5 - 2) x 180 = 540 degrees.
  • Exterior angle: Each exterior angle is 72 degrees (360/5).

Real-World Applications

  • Architecture: The Pentagon building in Washington, D.C. is one of the most famous pentagonal structures.
  • Nature: Many flowers have five petals arranged in a pentagonal pattern.
  • Sports: Soccer balls use pentagons and hexagons (truncated icosahedron).
  • Tiling: While regular pentagons cannot tile the plane alone, certain irregular pentagons can.