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Pentagon Calculator

Calculate regular pentagon area, perimeter, apothem, diagonals, and interior angles from side length with step-by-step math.

Regular Pentagon Dimensions
Area
43.0119
Perimeter
25
Apothem
3.441
Interior Angle
108°
Deduction Steps:
Regular polygon with n = 5 sides, side length s = 5
Perimeter: P = n × s = 5 × 5 = 25
Interior Angle: ((n - 2) × 180°) / n = ((5 - 2) × 180) / 5 = 108°
Exterior Angle: 360° / n = 360 / 5 = 72°
Apothem (inradius): a = s / (2 · tan(π/n)) = 3.441
Circumradius: R = s / (2 · sin(π/n)) = 4.2533
Area: A = ½ × P × a = 0.5 × 25 × 3.441 = 43.0119
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What Is a Pentagon Calculator?

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.

How to Use This Calculator

  1. Enter the side length (s) of the regular pentagon.
  2. View the Area, Perimeter, Apothem, and Interior Angles.
  3. Follow the step-by-step geometric derivations.

Regular Pentagon Area & Apothem Formulas

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.

Worked Example

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

Result: Area ≈ 61.94 cm², Perimeter = 30 cm, Apothem ≈ 4.13 cm

Tips & Key Notes

  • The interior angle of every regular pentagon is exactly 108°, and exterior angle is 72°.
  • The ratio of a pentagon’s diagonal to its side equals the golden ratio: d / s = (1 + √5)/2 ≈ 1.618034.
  • The sum of all interior angles of a pentagon is (5 - 2) × 180° = 540°.

Frequently Asked Questions

What is the apothem of a pentagon?

The apothem is the perpendicular distance from the center of the pentagon to the midpoint of any side.

What is the sum of interior angles in a pentagon?

The sum is always (5 - 2) × 180° = 540°. In a regular pentagon, each of the 5 angles is 540° / 5 = 108°.

How is the golden ratio related to a pentagon?

The diagonal of a regular pentagon divided by its side length equals the golden ratio φ = (1 + √5)/2 ≈ 1.618.

How do you calculate pentagon area using apothem?

Area = ½ × Perimeter × Apothem = ½ × 5s × a.

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