The Pentagon Area Calculator finds every key measurement of a regular pentagon from whichever single value you already know. Most geometry tools only go one direction (side in, area out), but this calculator solves in any direction: give it the area, the perimeter, the apothem (inradius) or the circumradius and it back-calculates the side length, then reports all six quantities — area, side, perimeter, inradius, circumradius and diagonal — together. A live SVG diagram labels the apothem and circumradius, and a collapsible “Show working” section walks through every algebraic step.
Everything runs entirely in your browser — no numbers are ever sent to a server.
How the maths works
A regular pentagon has five equal sides and five equal interior angles of 108°. All its geometry can be derived from a single measurement using the following relationships, where s is the side length and pi is approximately 3.14159:
| Property | Formula |
|---|---|
| Area (A) | (5 times s-squared) / (4 times tan(pi/5)) |
| Perimeter (P) | 5 times s |
| Inradius / apothem (a) | s / (2 times tan(pi/5)) |
| Circumradius (R) | s / (2 times sin(pi/5)) |
| Diagonal (d) | s times phi, where phi = (1 + sqrt(5)) / 2 |
The angle pi/5 = 36° appears because a regular pentagon’s central angle (from centre to two adjacent vertices) is 360°/5 = 72°, and the half-angle used in the inradius formula is 72°/2 = 36°. The value tan(36°) ≈ 0.72654 is the critical constant that links the side length to both the area and the apothem.
Inverse formulae — how the calculator back-calculates the side when you enter a different quantity:
| Known quantity | Formula for side s |
|---|---|
| Perimeter P | s = P / 5 |
| Inradius a | s = 2 times a times tan(pi/5) |
| Circumradius R | s = 2 times R times sin(pi/5) |
| Area A | s = sqrt(4 times A times tan(pi/5) / 5) |
Worked example
A regular pentagon has a side length of 10 cm. What are its area and other measurements?
- Area: A = (5 times 10-squared) / (4 times tan(36°)) = 500 / (4 times 0.72654) = 500 / 2.90617 ≈ 172.048 cm²
- Perimeter: P = 5 times 10 = 50 cm
- Inradius (apothem): a = 10 / (2 times 0.72654) = 10 / 1.45309 ≈ 6.882 cm
- Circumradius: R = 10 / (2 times sin(36°)) = 10 / (2 times 0.58779) ≈ 8.507 cm
- Diagonal: d = 10 times 1.61803 ≈ 16.180 cm (exactly 10 times the golden ratio!)
As a quick reference table for different side lengths:
| Side (cm) | Area (cm²) | Perimeter (cm) | Apothem (cm) | Diagonal (cm) |
|---|---|---|---|---|
| 5 | 43.01 | 25.00 | 3.44 | 8.09 |
| 10 | 172.05 | 50.00 | 6.88 | 16.18 |
| 20 | 688.19 | 100.00 | 13.76 | 32.36 |
| 50 | 4301.19 | 250.00 | 34.41 | 80.90 |
Notice that area scales with the square of the side length — doubling the side quadruples the area.
The golden ratio connection
One of the most remarkable facts in geometry is that the diagonal of a regular pentagon is exactly phi times its side length, where phi = (1 + sqrt(5)) / 2 ≈ 1.61803… is the golden ratio. This is not an approximation — it is an exact algebraic relationship. The golden ratio appears here because cos(36°) = phi/2 and the diagonals of a regular pentagon intersect each other in the golden ratio. This is why the five-pointed star (pentagram), formed by extending the sides of a regular pentagon, contains golden-ratio proportions at every turn.
Practical uses
Architecture and design. Pentagonal floor plans, garden beds, paving features and decorative tiles all require accurate area calculations for material estimates.
Engineering. Pentagonal cross-sections and bolt-hole patterns (some metric flanges use 5-bolt patterns) need precise inradius and circumradius values for machining.
Education. The pentagon is a favourite shape in school geometry for demonstrating the golden ratio and for exercises in trigonometric problem-solving.
Art and crafts. Origami pentagons, stained glass and quilting patterns start from a known side length; knowing the diagonal and circumradius makes cutting and assembly far easier.
A note on precision
The formulae used here are exact. The only rounding occurs in the displayed output, controlled by the decimal-places selector (default 6). For practical measurements — a tape measure reads to about ±1 mm — two or three decimal places is usually all that is meaningful. The full double-precision intermediate arithmetic (about 15-16 significant digits) ensures the displayed result is always correctly rounded.