Pentagon Calculator
Find all measurements of a regular pentagon instantly. Enter the side length and get area, perimeter, and diagonal based on the golden ratio.
Regular Pentagon Properties
A regular pentagon has five equal sides and five equal interior angles of 108 degrees each. The sum of interior angles in any pentagon is 540 degrees, so dividing by 5 gives 108 degrees per angle. This geometry creates a distinctive five-fold symmetry found throughout nature and design.
The pentagon's most fascinating property is its intimate relationship with the golden ratio φ = (1 + √5)/2 ≈ 1.618. The ratio of any diagonal to any side equals exactly φ. This mathematical constant appears in art, architecture, and natural growth patterns, making the pentagon a bridge between geometry and aesthetics.
You can divide a regular pentagon into five isosceles triangles meeting at the center. Each triangle has two equal sides (radii of the circumscribed circle) and a base equal to the pentagon's side. The apex angle of each triangle is 72 degrees (360°/5), and the base angles are 54 degrees each.
The Pentagon and the Golden Ratio
Draw all five diagonals inside a regular pentagon and they form a smaller inverted pentagon in the center. The ratio of the outer pentagon's side to the inner pentagon's side equals φ². This self-similar nesting pattern can continue infinitely, each layer related by the golden ratio.
Ancient Greek mathematicians discovered these properties and considered the pentagon sacred geometry. The pentagram (five-pointed star formed by extending the pentagon's sides) was a symbol of the Pythagorean school, representing the harmony of mathematical proportions in the universe.
Modern applications leverage the pentagon's golden ratio properties in design. The Pentagon building's shape was chosen partly for its symbolic strength and efficiency. Pentagons tile less efficiently than hexagons but create more visually interesting patterns, making them popular in flooring, quilting, and graphic design where aesthetics matter as much as function.
Calculating Pentagon Dimensions
The area formula A = (1/4)√(25 + 10√5) s² looks intimidating but reduces to approximately 1.720 s². This means a pentagon with 10 cm sides has an area of about 172 square centimeters. The formula comes from summing the areas of five isosceles triangles that compose the pentagon.
The diagonal length equals the side length times the golden ratio: d = s × φ ≈ 1.618s. A pentagon with 5-inch sides has diagonals of about 8.09 inches. This constant ratio makes pentagon construction straightforward: once you set the side length, all other dimensions follow from φ.
To find the circumradius (radius of the circle passing through all vertices), use R = s/(2 sin(36°)) ≈ 0.851s. The inradius (radius of the circle tangent to all sides) is r = s/(2 tan(36°)) ≈ 0.688s. These relationships help when inscribing or circumscribing pentagons in circles, a common task in compass-and-straightedge constructions and CAD design.
Frequently Asked Questions
What is a regular pentagon?
A regular pentagon is a five-sided polygon where all sides have equal length and all interior angles measure 108 degrees. The Pentagon building in Washington D.C. is the most famous example.
What is the formula for pentagon area?
For a regular pentagon with side length s, the area is A = (1/4)√(25 + 10√5) s², which equals approximately 1.720 s². This complex formula arises from the pentagon's connection to the golden ratio.
How do I find the perimeter of a pentagon?
The perimeter of a regular pentagon is P = 5s, where s is the side length. Since all five sides are equal, simply multiply the side length by 5.
What is the golden ratio in a pentagon?
The ratio of a pentagon's diagonal to its side length equals the golden ratio φ (phi), approximately 1.618. This means diagonal = s × φ. The pentagon is the simplest polygon that embeds the golden ratio geometrically.
Where do pentagons appear in nature?
Many flowers have five petals (pentagonal symmetry), including roses, buttercups, and apple blossoms. Starfish and sea urchins exhibit five-fold radial symmetry. The pentagon's connection to the golden ratio appears throughout biological growth patterns.