Calculate regular pentagon area, perimeter, apothem, diagonals, and interior angles from side length with step-by-step math.
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A regular pentagon is a five-sided polygon with all sides equal in length and all interior angles equal to 108°. It exhibits five lines of reflective symmetry and rotational symmetry of order 5.
A = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})} s^2 \approx 1.72048 s^2, \quad a = \frac{s}{2\tan(36^\circ)}Area relates to side length squared using exact radical geometry involving the golden ratio (φ). Apothem is the inradius.
Scenario: Find area and perimeter of a regular pentagon with side s = 6 cm
Perimeter: P = 5 × 6 = 30 cm
Apothem: a = 6 / (2 × tan(36°)) ≈ 4.1291 cm
Area: A = ½ × P × a = 0.5 × 30 × 4.1291 ≈ 61.9372 cm²
The apothem is the perpendicular distance from the center of the pentagon to the midpoint of any side.
The sum is always (5 - 2) × 180° = 540°. In a regular pentagon, each of the 5 angles is 540° / 5 = 108°.
The diagonal of a regular pentagon divided by its side length equals the golden ratio φ = (1 + √5)/2 ≈ 1.618.
Area = ½ × Perimeter × Apothem = ½ × 5s × a.
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