# 2D shapes – Pentagon

The pentagon is a five-sided polygon with five angles, and its sides are all the same length. If we let “s” represent the length of each side, we can use the following formulas to find some other important measurements of a regular pentagon:

• Perimeter: P = 5s
• Interior angle: θ = (3π – 2π/n)/2 = 108°
• Exterior angle: β = 360°/n = 72°
• Apothem: a = s/(2tan(π/n))
• Area: A = (1/2)Pa = (1/4)s²(5+5√5)

Here, “n” represents the number of sides in the polygon (in this case, n=5). The interior angle formula comes from the fact that the sum of the interior angles of a polygon with n sides is (n-2) × 180°. The apothem formula gives us the distance from the center of the pentagon to the midpoint of one of its sides.

The pentagon is one of the five Platonic solids, a group of regular, convex polyhedra that were first described by the ancient Greek philosopher Plato. The pentagon represents the element of “fire” in Plato’s system of cosmology.

In ancient Greece, the pentagon was also associated with the Pythagoreans, a mystical brotherhood of mathematicians and philosophers who believed that the pentagon had magical properties. They believed that the pentagon could be used to create perfect proportions in art and architecture.

The pentagon has been used as a design element in many cultures throughout history, from the intricate Islamic tile work of the Middle East to the traditional woven baskets of Native American tribes.