How do you find the area of a regular pentagon?

Area = (1/4) × √(5(5 + 2√5)) × side², which is roughly 1.7205 × side². This works for any regular five-sided polygon.

What is the diagonal of a regular pentagon?

The diagonal equals the side length times the golden ratio φ (about 1.618), so diagonal = side × 1.618.

What inputs can I start from?

You can enter any one of five quantities — side length, diagonal, circumradius, inradius (apothem) or area — and the tool derives the side, then every other property from it.

What is the apothem (inradius) of a pentagon?

The inradius or apothem is the distance from the centre to the middle of a side. For a regular pentagon it equals side ÷ (2·tan(36°)), about 0.688 × side.

What is the circumradius of a pentagon?

The circumradius is the distance from the centre to a vertex, equal to side ÷ (2·sin(36°)), about 0.851 × side. It is the radius of the circle that passes through all five corners.

What are the interior angles of a regular pentagon?

Each interior angle of a regular pentagon is 108°, and the five together sum to 540°. The central angle subtended by each side is 72°.

Does this tool upload anything?

No — all maths happens in your browser and no data is sent anywhere.

Pentagon Calculator — Gera Tools

Regular pentagon calculator

Tell the tool what you know — the side length, a diagonal, the circumradius, the inradius (apothem) or the area — and it derives every other property of a regular pentagon: area, perimeter, diagonal, circumradius and inradius. It is useful for geometry problems, tiling, drafting and design work involving five-sided shapes.

How it works

The calculator first converts your input into the side length, then applies the fixed ratios of a regular pentagon (φ is the golden ratio, (1+√5)/2 ≈ 1.618):

diagonal      = side × φ
circumradius  = side ÷ (2·sin 36°)   ≈ 0.851 × side
inradius      = side ÷ (2·tan 36°)   ≈ 0.688 × side
area          = (1/4)·√(5(5+2√5))·side²  ≈ 1.7205 × side²
perimeter     = 5 × side

If you enter the area instead of the side, it inverts the area formula (side = √(area ÷ 1.7205)) and the diagonal, radii and perimeter follow.

Example

A regular pentagon with side = 10:

Perimeter: 5 × 10 = 50
Diagonal: 10 × 1.618 = 16.18
Circumradius: 10 × 0.851 = 8.51
Inradius (apothem): 10 × 0.688 = 6.88
Area: 1.7205 × 100 = 172.05

Property	Multiple of side	Side 10
Diagonal	1.618	16.18
Circumradius	0.851	8.51
Inradius	0.688	6.88
Area	1.7205 × side²	172.05

All calculations stay in your browser.