Icosahedron Volume Calculator – Regular Polyhedron Properties
A regular icosahedron is one of the five classical Platonic solids and the one with the greatest number of faces. It has 20 equilateral triangular faces, 30 edges of equal length, and 12 vertices. Its geometry is deeply connected to the golden ratio φ = (1 + √5) / 2 ≈ 1.618, which appears in its volume, surface area, and all three characteristic sphere radii. This calculator computes every key property of a regular icosahedron from a single edge length input.
Volume Formula
For a regular icosahedron with edge length a, the volume is:
V = (5 / 12) × (3 + √5) × a³
Numerically, the constant evaluates to approximately 2.18169521 × a³. The formula arises from decomposing the icosahedron into pyramids, each with an equilateral triangular base and an apex at the centre of the solid. There are 20 such pyramids — one per face — and their combined volume gives the result above.
Example: if a = 5 cm, then V = (5/12) × (3 + √5) × 5³ ≈ 272.71 cm³.
Surface Area Formula
The total surface area of all twenty equilateral triangular faces is:
A = 5√3 × a²
Each equilateral triangular face has area (√3 / 4) × a², and there are 20 faces, so A = 20 × (√3 / 4) × a² = 5√3 × a². For a = 5 cm, the surface area is approximately 433.013 cm².
Characteristic Radii and the Golden Ratio
The icosahedron's geometry is governed by the golden ratio φ. All three characteristic spheres have radii that depend on φ directly:
- Inradius (r = aφ² / (2√3)) — the radius of the inscribed sphere (insphere). This sphere sits inside the icosahedron touching the centre of every face. For
a = 5 cm,r ≈ 3.786 cm. - Midradius (ρ = aφ / 2) — the radius of the midsphere. This sphere passes through the midpoint of every one of the 30 edges. For
a = 5 cm,ρ ≈ 4.045 cm. - Circumradius (R = a√(10 + 2√5) / 4) — the radius of the circumscribed sphere (circumsphere). This sphere passes through all 12 vertices. For
a = 5 cm,R ≈ 4.760 cm.
The ratio R / r ≈ 1.258 and ρ / r ≈ 1.068 are both determined solely by φ, reflecting the icosahedron's deep golden-ratio symmetry.
Supported Units
The calculator accepts edge lengths in millimetres (mm), centimetres (cm), metres (m), kilometres (km), inches (in), feet (ft), and yards (yd). The volume is expressed in the corresponding cubic unit and the surface area in the corresponding square unit. All radii share the same linear unit as the input. No cross-unit conversion is applied internally — select the unit that matches your measurement.
How to Use This Calculator
- Enter the edge length of the regular icosahedron in the input field. Any positive real number is valid.
- Select the unit from the dropdown — millimetres, centimetres, metres, kilometres, inches, feet, or yards.
- Results appear instantly, showing volume, surface area, inradius, midradius, and circumradius.
- Click the copy icon next to any result to copy that individual value, or use Copy All to copy the complete result set to your clipboard.
- Click Reset to restore the default example values.
Applications
Icosahedra appear in many scientific and design contexts. In virology, many viruses — including common cold viruses and adenoviruses — have icosahedral capsids, because the icosahedron is the most efficient symmetric shell for enclosing volume with equidistant protein subunits. In game design, the twenty-sided die (d20) used in tabletop role-playing games is an icosahedron. In architecture and engineering, geodesic domes are based on icosahedral geometry. Volume and surface area calculations help estimate material quantities, packing densities, and structural load distributions for icosahedral forms.
Accuracy and Limitations
All calculations use JavaScript's double-precision floating-point arithmetic (IEEE 754), providing approximately 15 significant digits of precision. Displayed results are rounded to 4 significant figures for readability; the copy-to-clipboard value provides 6 decimal places for greater precision. The calculator is valid for any positive edge length and assumes a perfect regular icosahedron — all 30 edges equal, all 20 faces equilateral. Truncated, elongated, or irregular icosahedra require different formulas not covered here.