Case 1
- Side length
- 6
61.937186
Open with these valuesPentagon Area Calculator
61.937186units²
Result: 61.937186 units²Five identical triangles meet at the centre, which is why the area is ¼√(25+10√5) times the side squared — about 1.720 s². A side of 6 gives 61.94 square units. Enter the side, not the distance from the centre to a corner.
Worked examples
Case 2
- Side length
- 1
1.720477
Open with these valuesCase 3
- Side length
- 10
172.047740
Open with these valuesHow it's calculated
A = ¼ × √(25 + 10√5) × s²
- StepMeasure one side — in a regular pentagon all five are equal.
- StepEnter it in any length unit; do not enter a radius here.
- ResultRead the area in the square of that unit.
Reference table
| Side | Perimeter, apothem, circumradius | Area |
|---|---|---|
| 0.5 | 2.5, 0.344095, 0.425325 | 0.430119 |
| 1 | 5, 0.688191, 0.850651 | 1.720477 |
| 2.5 | 12.5, 1.720477, 2.126627 | 10.752984 |
| 6 | 30, 4.129146, 5.103905 | 61.937186 |
| 10 | 50, 6.881910, 8.506508 | 172.047740 |
Questions
How do I calculate the area of a regular pentagon?
Multiply ¼ × √(25 + 10√5) — about 1.720 — by the side length squared. For a side of 6 that is 1.720 × 36 ≈ 61.937186 square units. The constant comes from splitting the pentagon into five identical isosceles triangles.
Do I enter the side or a radius?
The side, meaning one of the five equal edges. A pentagon with side 1 has a circumradius of only 0.850651, so entering that radius instead would give an area about a quarter too small.
What is the difference between the apothem and the circumradius?
The apothem runs from the centre to the middle of a side, s ÷ (2·tan36°), and is the radius of the largest circle that fits inside. The circumradius runs from the centre to a corner, s ÷ (2·sin36°), the radius of the smallest circle around it. For a side of 6 they are about 4.129146 and 5.103905.
Why does the pentagon formula use 36°?
Each side subtends a central angle of 360° ÷ 5 = 72°, and splitting that triangle down the middle leaves a right triangle with 36° at the centre. Half the side lies opposite that angle and the apothem beside it, which is where s ÷ (2·tan36°) and s ÷ (2·sin36°) come from.
What units does the pentagon calculator use?
Any length unit you like. Enter the side in centimetres and the area comes back in square centimetres; enter it in inches and you get square inches.
Sources and last check
Information, not professional advice.