Platonic Solids Calculator | Volume, Surface Area & Radii for All 5 Solids
Compute volume, surface area, inradius (insphere), midradius (midsphere), and circumradius (circumsphere) for all five Platonic solids — tetrahedron, cube, octahedron, dodecahedron, and icosahedron — from a single edge length. Includes face, edge, and vertex counts.
What Is the Platonic Solids Calculator | Volume, Surface Area & Radii for All 5 Solids?
The five Platonic solids are the only convex polyhedra whose faces are congruent regular polygons meeting at each vertex with the same configuration. They are the tetrahedron (4 triangular faces), cube (6 square faces), octahedron (8 triangular faces), dodecahedron (12 pentagonal faces), and icosahedron (20 triangular faces). For a solid with edge length a, all geometric properties — volume, surface area, inradius, midradius, and circumradius — can be expressed in closed form. The golden ratio φ = (1+√5)/2 appears in the dodecahedron and icosahedron formulas.
Formula
Tetrahedron: V = a³/(6√2), SA = a²√3 · Cube: V = a³, SA = 6a² · Octahedron: V = a³√2/3, SA = 2a²√3 · Dodecahedron: V = a³(15+7√5)/4 · Icosahedron: V = 5a³(3+√5)/12
How to Use
- 1
Enter the edge length a — this is the length of any one edge (all edges are equal for Platonic solids).
- 2
Choose a unit from the dropdown: mm, cm, m, in, or ft.
- 3
Optionally click a solid name button to pre-select which one to highlight in the detail card.
- 4
Click "Calculate All Solids" to compute volume, surface area, inradius, midradius, and circumradius for all five.
- 5
Read the highlighted solid card showing all properties and the dihedral angle.
- 6
Scroll to the comparison table — all five solids are shown side by side. Click any row name to switch the detail card.
- 7
Use the "Unit edge (a=1)" preset to compare normalized values without units.
Enter a single edge length in any unit (mm, cm, m, in, ft), optionally select a solid to highlight in detail, then click Calculate All Solids. A comparison table shows all five solids simultaneously. Click any row to view that solid's detailed card.
Example Calculation
For a cube with edge a = 10 cm: V = 10³ = 1000 cm³, SA = 6×10² = 600 cm², r_in = 5 cm, r_mid = 7.071 cm, r_circ = 8.660 cm. For a tetrahedron with a = 10 cm: V = 1000/(6√2) ≈ 117.9 cm³, SA = 100√3 ≈ 173.2 cm², r_circ = 10√6/4 ≈ 6.124 cm. The icosahedron packs the most volume for a given surface area among the five.
Understanding Platonic Solids | Volume, Surface Area & Radii for All 5 Solids
Properties of All Five Platonic Solids
| Solid | Faces | Face shape | Edges | Vertices | Dihedral angle | Dual solid |
|---|---|---|---|---|---|---|
| Tetrahedron | 4 | Equilateral triangle | 6 | 4 | 70.53° | Tetrahedron (self-dual) |
| Cube | 6 | Square | 12 | 8 | 90° | Octahedron |
| Octahedron | 8 | Equilateral triangle | 12 | 6 | 109.47° | Cube |
| Dodecahedron | 12 | Regular pentagon | 30 | 20 | 116.57° | Icosahedron |
| Icosahedron | 20 | Equilateral triangle | 30 | 12 | 138.19° | Dodecahedron |
Volume and Surface Area Formulas (edge length a)
| Solid | Volume V | Surface Area SA | Circumradius r_circ |
|---|---|---|---|
| Tetrahedron | a³ / (6√2) | a²√3 | a√6 / 4 |
| Cube | a³ | 6a² | a√3 / 2 |
| Octahedron | a³√2 / 3 | 2a²√3 | a / √2 |
| Dodecahedron | a³(15+7√5) / 4 | 3a²√(25+10√5) | a√3(1+√5) / 4 |
| Icosahedron | 5a³(3+√5) / 12 | 5a²√3 | a·sin(2π/5) |
Applications of Platonic Solids
- ›Crystallography: Crystal structures exhibit cubic (cube/octahedron) and tetrahedral symmetry. NaCl forms cubic crystals; diamond adopts a tetrahedral network.
- ›Board games and dice: Standard dice: d4 (tetrahedron), d6 (cube), d8 (octahedron), d12 (dodecahedron), d20 (icosahedron) — all five Platonic solids.
- ›Architecture: Geodesic domes use icosahedral symmetry. Buckminsterfullerene (C60) is a truncated icosahedron — related to the soccer ball pattern.
- ›Platonic solids in nature: Viruses often exhibit icosahedral symmetry for geometric efficiency. Some radiolarian microorganisms have dodecahedral skeletons.
- ›Kepler's Mysterium Cosmographicum: Kepler attempted to explain planetary orbits by nesting the five Platonic solids inside each other. Though incorrect, it drove him toward discovering true orbital laws.
Frequently Asked Questions
What are the five Platonic solids?
The five Platonic solids are: Tetrahedron (4 triangular faces, 4 vertices, 6 edges), Cube/Hexahedron (6 square faces, 8 vertices, 12 edges), Octahedron (8 triangular faces, 6 vertices, 12 edges), Dodecahedron (12 pentagonal faces, 20 vertices, 30 edges), and Icosahedron (20 triangular faces, 12 vertices, 30 edges). Plato associated them with the classical elements — tetrahedron with fire, cube with earth, octahedron with air, icosahedron with water, and dodecahedron with the cosmos.
What is Euler's characteristic formula for polyhedra?
Euler's formula states that for any convex polyhedron: V − E + F = 2, where V is the number of vertices, E is edges, and F is faces. This is the Euler characteristic χ = 2 for all five Platonic solids. Tetrahedron: 4−6+4=2; Cube: 8−12+6=2; Octahedron: 6−12+8=2; Dodecahedron: 20−30+12=2; Icosahedron: 12−30+20=2.
What is the difference between inradius, midradius, and circumradius?
The inradius (r_in) is the radius of the largest sphere that fits inside the solid, tangent to every face. The midradius (r_mid) is the radius of the sphere tangent to every edge midpoint. The circumradius (r_circ) is the radius of the smallest sphere that contains the solid, passing through every vertex. For a unit cube: r_in = 0.5, r_mid = 0.707, r_circ = 0.866.
Why does the golden ratio appear in Platonic solid formulas?
φ = (1+√5)/2 ≈ 1.618 appears in the dodecahedron and icosahedron because their geometry involves regular pentagons, and the ratio of diagonal to side in a regular pentagon equals φ. The dodecahedron midradius = a(1+√5)/4 = aφ/2, and the icosahedron circumradius involves φ through the coordinates of its vertices, which can be expressed as permutations of (0, ±1, ±φ).
Which Platonic solid is most efficient for packing or containment?
The isoperimetric efficiency (volume-to-surface-area ratio) increases with the number of faces. The icosahedron (20 faces) has the highest ratio among Platonic solids — it is closest to a sphere. The tetrahedron (4 faces) has the lowest ratio. For space-filling, the cube is the only Platonic solid that tessellates 3D space without gaps, making it most relevant for construction and packing problems.
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