Regular pentagon area calculator

What is a regular pentagon area calculator?

A regular pentagon area calculator finds the area enclosed by a five-sided polygon whose sides are all the same length and whose interior angles are all equal to 108°. The only measurement you need is the side length — every other dimension (the apothem, the diagonal, the circumradius) is fixed by the geometry once the side is known.

This tool takes a single side length in any common unit and returns the area in the matching square unit. Switching the side or area unit reconverts the result automatically.

Key concepts

• Side length (s) — the length of one of the five equal edges of the pentagon.
• Apothem (a) — the perpendicular distance from the center of the pentagon to the midpoint of any side. For a regular pentagon, $a = \frac{s}{2 \tan(36°)}$.
• Interior angle — each of the five interior angles of a regular pentagon equals 108°.
• Golden ratio — the regular pentagon is famously linked to $\varphi = \frac{1 + \sqrt{5}}{2}$; the ratio of any diagonal to a side equals $\varphi$.

How does the calculator work?

The area of a regular pentagon depends on the square of the side length multiplied by a constant. That constant comes from splitting the pentagon into five congruent isosceles triangles meeting at the center, computing each triangle’s area, and summing them.

Formula

An equivalent apothem-based form, useful when you already know the apothem, is:

where $P = 5s$ is the perimeter and $a$ is the apothem.

Worked examples

Example 1: side = 10 cm

Example 2: side = 1

This is the dimensionless constant: the area of a unit-side regular pentagon.

Example 3: side = 5

Example 4: apothem-based check

For $s = 10$ cm the apothem is $a = \frac{10}{2 \tan(36°)} \approx 6.8819$ cm, so

which matches Example 1.

Practical uses

• Architecture and design — laying out pentagonal floors, tiles, gazebos, or windows.
• Engineering — sizing pentagonal cross-sections of bolts, nuts, and structural members.
• Cartography and planning — estimating the footprint of pentagonal plots or buildings (the Pentagon in Arlington is the most famous example).
• Mathematics and education — illustrating the golden ratio, demonstrating that regular polygons have closed-form areas, and comparing with the regular polygon area calculator for general $n$.

Notes

• The side length must be positive for the result to be meaningful; a zero side yields zero area.
• Units of side and area match: a side in metres gives an area in square metres. Switching the unit selectors reconverts the result automatically.
• For other regular polygons, see the regular hexagon area calculator and the regular octagon area calculator.