Pentagon Calculator

How to Calculate Regular Pentagon Geometry

A regular pentagon is a five-sided polygon with equal side lengths and equal interior angles of exactly 108°. Because of its symmetry, a single known measurement is enough to compute all other geometric properties.

Key Regular Pentagon Formulas

Using the side length s, the primary formulas are:

• Area (A): A = 0.25 * √(5 * (5 + 2√5)) * s² ≈ 1.720477 * s²
• Perimeter (P): P = 5 * s
• Apothem (a): a = s / (2 * tan(36°)) ≈ 0.688191 * s
• Circumradius (R): R = s / (2 * sin(36°)) ≈ 0.850651 * s
• Diagonal (d): d = s * φ ≈ 1.618034 * s (where φ is the Golden Ratio)

Step-by-Step Example

Suppose you have a pentagonal paving stone with a side length of 4 cm:

1. Side (s): 4 cm
2. Perimeter (P): 5 * 4 = 20 cm
3. Area (A): 1.720477 * 4² = 1.720477 * 16 ≈ 27.528 cm²
4. Apothem (a): 4 * 0.688191 ≈ 2.753 cm (this is the distance from the center to the flat edge)
5. Diagonal (d): 4 * 1.618034 ≈ 6.472 cm

Also check: Hexagon Calculator, Octagon Calculator, Regular Polygon Calculator, Pentagonal Prism Calculator, Golden Ratio Calculator, Area Calculator.