Pentagon Calculator — Properties of a Regular Pentagon
Welcome to the pentagon calculator, a tool that helps you find all essential properties of a regular pentagon. Enter any side length, choose your preferred unit (metric or imperial), and instantly get the area, perimeter, diagonal, height, circumradius, and inradius.
What is a pentagon? How many sides does a pentagon have?
A pentagon is a polygon with exactly 5 sides. It comes in many forms — convex, concave, or self-intersecting — but the most studied is the regular simple pentagon, where all five sides are equal in length and all five interior angles are equal.
The sum of the interior angles of any simple pentagon is 540°. In a regular pentagon, each interior angle equals 540° ÷ 5 = 108°.
Area and perimeter of a regular pentagon
The perimeter of a regular pentagon is straightforward — just multiply the side length by the number of sides:
P = 5 × a
where a is the side length.
The area of a regular pentagon is:
A = (a² × √(25 + 10√5)) / 4
This formula is derived from dividing the pentagon into 5 congruent isosceles triangles radiating from the center. Numerically, A ≈ 1.7205 × a².
Alternatively, using the circumscribed circle radius R:
A = (5 × R² × √((5 + √5) / 2)) / 4
How do I find the height and diagonal of a pentagon?
The diagonal of a regular pentagon connects two non-adjacent vertices. Remarkably, the ratio of the diagonal to the side equals the golden ratio φ ≈ 1.618:
d = a × φ = a × (1 + √5) / 2
The height of a regular pentagon is the perpendicular distance from the midpoint of one side to the opposite vertex. It equals the sum of the inradius and the circumradius:
h = R + r
where R is the circumradius and r is the inradius (apothem).
Circumradius and inradius of a regular pentagon
Two radii fully characterize a regular pentagon:
- Circumradius (R) — radius of the circumscribed circle (passes through all 5 vertices):
R = a / (2 × sin 36°) ≈ 0.8507 × a - Inradius or apothem (r) — radius of the inscribed circle (tangent to all 5 sides):
r = a / (2 × tan 36°) ≈ 0.6882 × a
How to solve a regular pentagon using this pentagon calculator?
Simply enter the side length (a) and select a unit — metric (mm, cm, m, km) or imperial (in, ft, yd, mi). The calculator instantly computes:
- Side (a) — the input value
- Perimeter (P) — total boundary length
- Area (A) — enclosed surface area
- Diagonal (d) — distance between non-adjacent vertices
- Height (h) — perpendicular distance from midpoint of a side to opposite vertex
- Circumradius (R) — radius of circumscribed circle
- Inradius / Apothem (r) — radius of inscribed circle
- Interior angle — always 108°
Other regular shapes
The pentagon belongs to the family of regular polygons. Here is a comparison of key properties:
- Triangle (3 sides) — interior angle 60°; area = (√3/4) × a²
- Square (4 sides) — interior angle 90°; area = a²
- Pentagon (5 sides) — interior angle 108°; area ≈ 1.7205 × a²
- Hexagon (6 sides) — interior angle 120°; area = (3√3/2) × a²
- Octagon (8 sides) — interior angle 135°; area ≈ 4.8284 × a²
As the number of sides increases, a regular polygon approaches a circle.
The pentagon and the golden ratio
One of the most fascinating properties of the regular pentagon is its deep connection to the golden ratio φ = (1 + √5) / 2 ≈ 1.618. The diagonal-to-side ratio equals φ exactly. Furthermore, the diagonals of a regular pentagon intersect each other in the golden ratio, creating the famous pentagram — a five-pointed star formed by connecting every other vertex.
Pentagon in architecture and nature
- The Pentagon (USA) — the headquarters of the U.S. Department of Defense is a regular pentagonal building in Arlington, Virginia.
- Starfish — most starfish have five-fold (pentagonal) radial symmetry.
- Flowers — many flower petals (roses, morning glories) display pentagonal symmetry.
- Viruses — many viral capsids (e.g., adenoviruses) have icosahedral symmetry, which incorporates pentagonal faces.
- Soccer balls — a standard soccer ball uses a pattern of 12 regular pentagons and 20 hexagons.
FAQs
Q: What is the area of a regular pentagon with side 10 cm?
A: A = (10² × √(25 + 10√5)) / 4 ≈ (100 × 8.8989) / 4 ≈ 172.05 cm²
Q: What is the perimeter of a regular pentagon with side 6 inches?
A: P = 5 × 6 = 30 inches
Q: How many diagonals does a pentagon have?
A: A regular pentagon has 5 diagonals. They form a pentagram inside the pentagon.
Q: Is the diagonal of a pentagon related to the golden ratio?
A: Yes! The ratio of a diagonal to a side equals the golden ratio φ ≈ 1.618034.
Q: What is the interior angle of a pentagon?
A: Each interior angle of a regular pentagon is exactly 108°. The sum of all interior angles is 540°.