Case 1
- Edge length (not the circumradius)
- 3
58.905765
Open with these valuesRegular Icosahedron Volume Calculator
58.905765units³
Result: 58.905765 units³All thirty edges of a regular icosahedron are equal, so one edge fixes the whole solid: V = 5(3 + √5) ÷ 12 × a³, a factor of about 2.181695. An edge of 1 holds 2.181695 cubic units, an edge of 3 holds 58.905765. The surface area 5√3 × a² stands in the table below.
Worked examples
Case 2
- Edge length (not the circumradius)
- 1
2.181695
Open with these valuesCase 3
- Edge length (not the circumradius)
- 5
272.711874
Open with these valuesHow it's calculated
V = 5(3 + √5) ÷ 12 × a³
- StepMeasure one edge — not the radius of the sphere around the solid.
- StepCube it and multiply by 5(3 + √5) ÷ 12 ≈ 2.181695.
- ResultRead the volume in the cube of the unit you entered.
Reference table
| Edge a | Exact volume | Surface area | Volume |
|---|---|---|---|
| 1 | 5(3 + √5) ÷ 12 | 8.660254 | 2.181695 |
| 2 | 5(3 + √5) × 8 ÷ 12 | 34.641016 | 17.453560 |
| 2.5 | 5(3 + √5) × 15.625 ÷ 12 | 54.126588 | 34.088984 |
| 3 | 5(3 + √5) × 27 ÷ 12 | 77.942286 | 58.905765 |
| 5 | 5(3 + √5) × 125 ÷ 12 | 216.506351 | 272.711874 |
Questions
How do I calculate the volume of a regular icosahedron?
Cube the edge length and multiply by 5(3 + √5) ÷ 12, a factor of about 2.181695. For an edge of 3 that is 2.181695 × 27 ≈ 58.905765 cubic units.
What is a regular icosahedron?
It is one of the five Platonic solids: twenty identical equilateral triangles meeting at twelve vertices, with thirty equal edges. Because every edge is the same length, a single edge measurement fixes the whole shape. A d20 gaming die is a regular icosahedron.
What is the surface area of a regular icosahedron?
It is the total of twenty equilateral triangles, 5 × √3 × a², and the table above lists it for every example edge. For an edge of 3 that is about 77.942286 square units. Surface area scales with the square of the edge, so doubling the edge quadruples it.
Do I enter the edge or the radius?
The edge, meaning the length of one of the thirty straight sides. The circumradius of an icosahedron is roughly 0.951 of its edge, and using it in this formula would give a badly wrong answer.
How does an icosahedron compare to a dodecahedron?
At the same edge length the dodecahedron holds far more: its factor is about 7.6631 against 2.1817. The icosahedron has more faces, twenty triangles against twelve pentagons, but they wrap tightly around a smaller volume.
Which units does the answer use?
Whatever you entered, cubed. An edge in centimetres gives cubic centimetres, an edge in inches cubic inches. The surface area in the table follows the same unit squared.
Sources and last check
Information, not professional advice.