Pentagon

This free online pentagon calculator helps you compute the area, perimeter, diagonal, height, circumradius, and apothem (inradius) of a regular pentagon from just its side length. Whether you are a student learning geometry, a teacher preparing lesson materials, a designer working with pentagonal shapes, an architect planning pentagonal structures, or a professional in engineering and construction, this tool provides instant and accurate results for all pentagon calculations.

The area of a regular pentagon is calculated using the formula: Area = (5/4) x a^2 x cot(pi/5), where a is the side length. This is equivalent to the standard formula Area = a^2 x sqrt(25 + 10sqrt5) / 4. The perimeter is simply 5 times the side length since all five sides of a regular pentagon are equal in length. A regular pentagon has interior angles of 108 deg each and exterior angles of 72 deg each, with the sum of all interior angles equaling 540 deg.

For more advanced calculations, the circumradius (radius of the circumscribed circle that passes through all five vertices) is R = a / (2 x sin(pi/5)), and the inradius or apothem (radius of the inscribed circle that touches all five sides) is r = a / (2 x tan(pi/5)). The height of a regular pentagon (distance between two parallel sides) is h = a x sqrt(5 + 2sqrt5) / 2. The diagonal length follows the golden ratio: d = a x (1 + sqrt5) / 2 ≈ a x 1.618, which is the same ratio found in the Parthenon, the Mona Lisa, and other classical works of art and architecture. A regular pentagon has five diagonals that form a pentagram (five-pointed star).

This calculator handles edge cases gracefully - side lengths of zero return zero results, negative values are rejected with validation, and very large inputs are handled without overflow. All results are formatted to two decimal places for clarity. The pentagram formed by drawing all diagonals of a regular pentagon has been a symbol used throughout history, from ancient Greek mathematics to modern-day geometry. The golden ratio φ = (1 + sqrt5) / 2 appears naturally in the pentagon's diagonal-to-side ratio, making the pentagon one of the most mathematically significant polygons in geometry.