Calculate the area, perimeter, apothem, circumradius, diagonal, and height of a regular pentagon from any single known dimension.
Pentagon Formula
A regular pentagon has 5 equal sides and 5 equal angles. Every property scales from the side length, so once you know the side you can find the rest. The core formulas are:
P = 5 * s
A = (1/4) * sqrt(25 + 10*sqrt(5)) * s^2
a = s / (2 * tan(36 deg))
R = s / (2 * sin(36 deg))
d = s * (1 + sqrt(5)) / 2
h = (1/2) * sqrt(5 + 2*sqrt(5)) * s
Where:
- s = length of one side
- P = perimeter (the distance around all 5 sides)
- A = area enclosed by the pentagon
- a = apothem, the distance from the center to the middle of a side (also called the inradius)
- R = circumradius, the distance from the center to a vertex
- d = diagonal, the distance from one vertex to a non-adjacent vertex
- h = height, the distance from one side to the opposite vertex
The calculator lets you start from any one of these values. If you enter the perimeter it divides by 5 to get the side; if you enter the area, apothem, circumradius, diagonal, or height, it works backward to the side first, then computes every other property. The area can also be written as A = (1/2) * P * a, which is half the perimeter times the apothem and applies to any regular polygon.
Regular Pentagon Property Ratios
Because each property is a fixed multiple of the side length, you can estimate any dimension by multiplying the side by the constant below. Each interior angle of a regular pentagon is always 108 degrees and each central angle is 72 degrees.
| Property | Value when side = s | Multiplier |
|---|---|---|
| Perimeter | 5s | 5.000000 |
| Area | 1.720477 s squared | 1.720477 |
| Apothem (inradius) | 0.688191 s | 0.688191 |
| Circumradius | 0.850651 s | 0.850651 |
| Diagonal | 1.618034 s | 1.618034 |
| Height | 1.538842 s | 1.538842 |
The diagonal multiplier 1.618034 is the golden ratio, which is why pentagons appear so often in discussions of that number.
Example Problems
Example 1: Area from the side. A regular pentagon has a side length of 6 cm. The area is A = 1.720477 * 6^2 = 1.720477 * 36 = 61.94 cm squared. The perimeter is 5 * 6 = 30 cm and the apothem is 0.688191 * 6 = 4.13 cm.
Example 2: Side from the area. A regular pentagon has an area of 100 in squared. Solve for the side: s = sqrt(100 / 1.720477) = sqrt(58.12) = 7.62 in. From there the perimeter is 5 * 7.62 = 38.12 in and the circumradius is 0.850651 * 7.62 = 6.49 in.
FAQ
What is the apothem of a pentagon? The apothem is the perpendicular distance from the center of the pentagon to the midpoint of any side. It is also called the inradius because it is the radius of the largest circle that fits inside the pentagon. For a regular pentagon, the apothem equals s / (2 * tan(36 degrees)), or about 0.688191 times the side length.
What is the difference between the apothem and the circumradius? The apothem reaches from the center to the middle of a side, while the circumradius reaches from the center to a vertex (corner). The circumradius is always larger. For a regular pentagon the circumradius is about 0.850651 times the side, compared to 0.688191 times the side for the apothem.
Do these formulas work for an irregular pentagon? No. These formulas assume a regular pentagon with 5 equal sides and 5 equal angles. For an irregular pentagon you would need to split the shape into triangles and add their areas, since there is no single side length that defines the whole figure.
