What is a regular pentagon?
A regular pentagon is a five-sided polygon in which all five sides have the same length and all five interior angles are equal. Its geometry is especially useful because the diagonal is connected to the golden ratio.
Five equal sides
The perimeter is simply five times the side length.
Five equal angles
Each interior angle is 108°, so the five angles total 540°.
How to use the Pentagon Calculator
Enter any one of the supported dimensions. The calculator determines the side length first, then derives every other regular-pentagon property from it.
Choose units
Select cm, mm, m, in or ft.
Enter one value
Side, area, perimeter, height, diagonal or radius.
Calculate
The regular-pentagon relationships determine the side.
Read the geometry
Area, perimeter, diagonal, height and radii update together.
Regular Pentagon Formulas
Let a be the side length. These are the core relationships used by the calculator.
Pentagon dimensions at a glance
| Symbol | Meaning | Multiplier of a | Practical relationship |
|---|---|---|---|
| a | Side | 1 | Base dimension |
| d | Diagonal | 1.618034 | d = φa |
| h | Height | 1.538842 | h = a√(5+2√5)/2 |
| R | Circumradius | 0.850651 | R = a/[2sin36°] |
| r | Apothem | 0.688191 | r = a/[2tan36°] |
| A | Area | 1.720477a² | A = a²√(25+10√5)/4 |
Why the diagonal uses φ
Draw all five diagonals of a regular pentagon and they form a pentagram. The intersections divide the diagonals into segments that repeat the same geometric proportion. That proportion is the golden ratio, φ = (1 + √5) / 2.
Worked Pentagon Examples
Side a = 10 cm
The perimeter is five sides:
The diagonal is:
The area is:
Area A = 172.048 cm²
Rearrange the area formula to recover the side:
For this example:
Perimeter P = 75 cm
Diagonal d = 20 cm
Important Pentagon Properties
- Number of sides: 5.
- Interior-angle sum: (5 − 2) × 180° = 540°.
- Each interior angle: 540° / 5 = 108°.
- Central angle: 360° / 5 = 72°.
- Symmetry: a regular pentagon has five lines of reflection symmetry and rotational symmetry of order 5.
- Golden ratio: the diagonal-to-side ratio is φ ≈ 1.618034.
- Area construction: the pentagon can be divided into five congruent isosceles triangles from its centre.
Common Pentagon Calculation Mistakes
Confusing height and apothem
The height runs between opposite parallel support lines of the regular pentagon. The apothem is the perpendicular distance from the centre to a side. They are different quantities.
Using the diagonal as the side
The diagonal is longer than the side by a factor of φ. If d is known, recover the side with a = d/φ before using the area formula.
Mixing area and length units
If the side is expressed in cm, area is in cm². Changing a length unit by a factor changes area by the square of that factor.
Using a non-regular pentagon formula
These compact formulas assume all five sides and all five angles are equal. An arbitrary five-sided polygon needs more information.
Pentagon Calculator FAQ
It is a five-sided polygon with five equal sides and five equal interior angles of 108°.
For side length a, A = a²√(25 + 10√5) / 4. Equivalently, area can be written as one-half of perimeter times apothem.
Multiply the side by the golden ratio: d = φa, where φ ≈ 1.618034.
The circumradius R is the distance from the centre to a vertex. The apothem r is the perpendicular distance from the centre to a side.
Yes. For a regular pentagon, a = P/5. The calculator then derives the other dimensions.
No. The formulas on this page are for a regular pentagon. An irregular pentagon may have unequal sides and angles.