Pentagon Calculator — Area, Perimeter & Diagonals
Find every dimension of a regular pentagon — area, perimeter, circumscribed circle radius, inscribed circle radius (apothem), and golden-ratio diagonals — from the side length, circumradius, or apothem.
Known measurement
Area = (5 × s²) / (4 × tan 36°) ≈ 1.7205 × s²
- 1
Side squared (s²)
5² = 25 - 2
5 × s²
5 × 25 = 125 - 3
Area = 5s² ÷ (4 × tan 36°)
125 ÷ 2.9062 = 43.01194 × tan(36°) ≈ 2.9062 is the denominator from the pentagon area formula.
How does this calculator work?
Regular pentagon: Area = (5 s²) / (4 tan 36°) ≈ 1.7205 s²; Perimeter = 5 s; Circumradius R ≈ 0.8507 s; Apothem r ≈ 0.6882 s; Diagonal = φ s ≈ 1.618 s (golden ratio); 5 diagonals; all interior angles = 108°. Enter any one measurement to get all others.
Formula
How this is calculated
A regular pentagon has 5 equal sides (length s) and 5 equal interior angles of 108° each. All its properties follow from the side length using the geometry of the isoceles triangles formed by the centre and each pair of adjacent vertices. The central angle is 360° / 5 = 72°, and the half-central triangle gives a right triangle with the angle 36° (π/5).
The circumradius (centre to vertex) is R = s / (2 sin 36°) ≈ 0.8507 s, and the inradius or apothem (centre to midpoint of a side) is r = s / (2 tan 36°) ≈ 0.6882 s. The area equals the perimeter times the apothem divided by 2: Area = (5s / 2) × r = (5 s²) / (4 tan 36°) ≈ 1.7205 s². A regular pentagon has exactly 5 diagonals, and each diagonal has length d = φ × s where φ = (1 + √5) / 2 ≈ 1.618 is the golden ratio — a famous geometric property.
You can enter any one of side length, circumradius, or inradius; the calculator derives the side from your input and then computes all other measurements in the same unit.
Frequently asked questions
Each interior angle is (5 − 2) × 180° / 5 = 108°. The exterior angle is 72°. All five interior angles are equal in a regular pentagon.
The diagonal d of a regular pentagon with side s satisfies d / s = φ = (1 + √5) / 2 ≈ 1.618, the golden ratio. This arises because the 36°-72°-72° isoceles triangle formed by the diagonal and two sides is a "golden gnomon", connecting the pentagon deeply to φ.
A regular pentagon has n(n − 3) / 2 = 5 × 2 / 2 = 5 diagonals. All five have the same length d = φ × s, where φ is the golden ratio.
Also known as
TG we-Calculate Editorial Team. (2026). Pentagon Calculator — Area, Perimeter & Diagonals [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pentagon-calculator
TG we-Calculate Editorial Team. "Pentagon Calculator — Area, Perimeter & Diagonals." TG we-Calculate. 2026. https://we-calculate.com/calculator/pentagon-calculator.
TG we-Calculate Editorial Team, "Pentagon Calculator — Area, Perimeter & Diagonals," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pentagon-calculator
@misc{wecalculate_pentagon_calculator, title = {Pentagon Calculator — Area, Perimeter & Diagonals}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pentagon-calculator}}, year = {2026}, note = {TG we-Calculate} }
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