Pentagon Calculator

A pentagon is a 2D polygon with exactly 5 sides and 5 angles. In a regular pentagon, all sides are equal in length and all interior angles are equal to 108°, with the sum of all interior angles being 540°.

The area of a regular pentagon is calculated using the formula: A = a² × √(25 + 10√5) / 4, where a is the side length. This simplifies to approximately A = 1.72048 × a². For example, a pentagon with side 5 has an area of about 43.01 square units.

The perimeter is simply 5 times the side length: P = 5 × a. Since all five sides of a regular pentagon are equal, you just multiply the side length by 5.

The diagonal of a regular pentagon connects two non-adjacent vertices. It is calculated as d = a × (1 + √5) / 2, which is the side length multiplied by the golden ratio (approximately 1.618). For a side of 5, the diagonal is about 8.09 units.

The height of a regular pentagon is the distance from one side to the opposite vertex. Use the formula: h = a × √(5 + 2√5) / 2. For a side length of 5, this gives a height of approximately 6.88 units.

The circumradius (R) is the radius of the circle that passes through all five vertices of the pentagon. It is calculated as R = a / (2 × sin(π/5)), which equals approximately 0.8507 × a.

The inradius or apothem (r) is the radius of the largest circle that fits inside the pentagon, touching all five sides. It is calculated as r = a / (2 × tan(π/5)), which equals approximately 0.6882 × a. The apothem is also useful for calculating the area: A = 0.5 × perimeter × apothem.

The apothem is the same as the inradius and represents the distance from the center of the pentagon to the midpoint of any side. Use the formula r = a / (2 × tan(36°)), or equivalently r ≈ 0.6882 × a, where a is the side length.