Calculate all properties of a regular pentagon including area, perimeter, side length, apothem, radius, and diagonal. Enter any known value for instant results.
A regular pentagon is a five-sided polygon where all sides are equal length and all interior angles are 108°. The pentagon is deeply connected to the golden ratio (φ ≈ 1.618), appearing in its diagonal-to-side ratio and making it a fundamental shape in art, architecture, and nature.
For a regular pentagon with side length s, the area formula is A = (1/4)√(25 + 10√5)s², which simplifies to approximately 1.720s². For example, if the side is 5 units, the area is about 1.720 × 25 ≈ 43.01 square units. This formula comes from dividing the pentagon into 5 triangles from the center.
The golden ratio (φ ≈ 1.618) appears naturally in regular pentagons. The ratio of a diagonal to a side equals φ. This means if your pentagon has a side of 5 units, the diagonal will be about 8.09 units (5 × 1.618). This mathematical beauty makes pentagons special in art and architecture.
A pentagon has 5 diagonals total. Each diagonal connects two non-adjacent vertices. From any vertex, you can draw 2 diagonals (since you can't connect to itself or the two adjacent vertices). With 5 vertices, that's 5 × 2 / 2 = 5 diagonals total. All diagonals in a regular pentagon have the same length.
The apothem is the distance from the center of the pentagon to the midpoint of any side. It can be calculated using the formula a = s/(2×tan(36°)), where s is the side length. The apothem is perpendicular to the side and represents the height of the triangular segments that make up the pentagon.
The interior angle formula for any regular polygon is 180°×(n-2)/n, where n is the number of sides. For a pentagon (n=5), this gives 180°×(5-2)/5 = 180°×3/5 = 108°. Since all five angles are equal in a regular pentagon, each measures exactly 108°, and all five angles sum to 540°.
Regular pentagons cannot tile a flat surface by themselves because their 108° interior angles don't divide evenly into 360°. However, irregular pentagons can tile surfaces, and there are 15 known types of convex pentagons that can tile a plane. This remains an active area of mathematical research.
The radius (circumradius) of a regular pentagon is the distance from the center to any vertex. It can be calculated using r = s/(2×sin(36°)), where s is the side length. Since sin(36°) ≈ 0.588, the radius is approximately 0.851 times the side length. For a 5-unit side, the radius is about 4.25 units.
The pentagon is unique because it's the first polygon where the diagonal-to-side ratio equals the golden ratio (φ). It's also the largest regular polygon that can be a face of a Platonic solid (the dodecahedron has 12 pentagonal faces). Pentagons appear frequently in nature, especially in flowers with 5-fold symmetry.
This calculator uses precise mathematical formulas and displays results to 4 decimal places. The calculations involve the golden ratio and trigonometric functions (sin and tan of 36°), which are computed with high precision. Results are accurate for all practical purposes including engineering, architecture, and scientific applications.
“Perfect for my geometry class! Understanding the golden ratio in pentagons has never been easier. The visual charts really help me grasp the relationships between different measurements.”
“As an architect, I use this for pentagon-based designs. The ability to input any measurement and get all others instantly saves me tons of time. Export feature is great for documentation!”
“Incredibly accurate and easy to use. I design sacred geometry art, and the golden ratio calculations are spot-on. The formulas displayed help me understand the mathematics behind the beauty.”