Pentagon calculator
This pentagon calculator works out a regular pentagon from one known value: side length, circumradius, inradius, area, perimeter, diagonal or height.
Example: Known value: Side length a · Value 5 · Unit: cm → Result: Side 5 cm: area 43.0119 cm², perimeter 25 cm. Source: Wolfram MathWorld, Regular Pentagon: circumradius R = (1/10)·√(50+10√5)·a, inradius r = (1/10)·√(25+10√5)·a, area A = (1/4)·√(25+10√5)·a², diagonal = φ·a with φ = (1+√5)/2 (read 14.10.2026). Updated: .
How it is calculated
The area of a regular pentagon is A = ¼·√(25 + 10√5)·a² ≈ 1.7205·a². With a side of 5 cm the area is about 43.0119 cm² and the perimeter 25 cm.
Pentagon formulas
- Perimeter: P = 5a
- Circumradius: R = (1/10)·√(50 + 10√5)·a ≈ 0.8507·a
- Inradius (apothem): r = (1/10)·√(25 + 10√5)·a ≈ 0.6882·a
- Diagonal: d = φ·a ≈ 1.618·a, where φ = (1 + √5)/2 is the golden ratio
- Height from a corner to the opposite side: h = R + r ≈ 1.5388·a
- Each interior angle is 108°, together 540°.
The area also follows from the general rule for regular polygons, A = ½·perimeter·apothem: for a = 5 that is ½ · 25 · 3.441 ≈ 43.01.
Examples
| Side a | Area | Perimeter | Circumradius R | Diagonal d |
|---|---|---|---|---|
| 1 | 1.7205 | 5 | 0.8507 | 1.618 |
| 2 | 6.8819 | 10 | 1.7013 | 3.2361 |
| 5 | 43.0119 | 25 | 4.2533 | 8.0902 |
| 10 | 172.0477 | 50 | 8.5065 | 16.1803 |
The calculator also runs backwards: from an area of 100 cm² it finds the side √(100 ÷ 1.7205) ≈ 7.6239 cm. For six sides use the hexagon calculator, for any number of sides the regular polygon calculator.
Limits
All formulas assume a regular pentagon. An irregular pentagon needs its corner coordinates or a split into triangles; this tool does not cover that case.
Frequently asked questions
How do you find the area of a pentagon?
For a regular pentagon: A = ¼·√(25 + 10√5)·a² ≈ 1.7205·a². A side of 10 gives about 172.0477.
What is the formula for the perimeter of a pentagon?
P = 5a. A pentagon with 5 cm sides has a perimeter of 25 cm.
How long is the diagonal of a pentagon?
The diagonal is the side times the golden ratio: d = φ·a ≈ 1.618·a.
What is the interior angle of a pentagon?
Each interior angle of a regular pentagon is 108°; the five angles add up to 540°.
How do I get the side from the area?
a = √(A ÷ 1.7205). Choose “Area” as the known value and the tool does it for you.
Sources and legal basis
- Wolfram MathWorld, Regular Pentagon: circumradius R = (1/10)·√(50+10√5)·a, inradius r = (1/10)·√(25+10√5)·a, area A = (1/4)·√(25+10√5)·a², diagonal = φ·a with φ = (1+√5)/2 (read 14.10.2026)
- Wolfram MathWorld, Regular Polygon: interior angle (n−2)·180°/n, area = n·a²·cot(π/n)/4 (read 14.10.2026)
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