GEOMETRY3D GeometryMathematics Calculator
🔺

Compute volume and surface area of a regular tetrahedron

Enter edge length to get volume (a³√2/12), surface area (√3·a²), height, inradius, and circumradius. Simplest Platonic solid.

Concept Fundamentals
V = a³√2/12
Volume
A = √3·a²
Surface Area
h = a√(2/3)
Height
4 equilateral triangles
4 faces

Did our AI summary help? Let us know.

Volume = a³√2/12 — derived from pyramid formula 4 equilateral faces, 6 equal edges One of 5 Platonic solids Height h = a√(2/3), inradius r = a/(2√6) Dihedral angle ~70.53°

Key quantities
V = a³√2/12
Volume
Key relation
A = √3·a²
Surface Area
Key relation
h = a√(2/3)
Height
Key relation
4 equilateral triangles
4 faces
Key relation

Ready to run the numbers?

Why: Tetrahedra appear in chemistry (CH₄, diamond), physics (FEM meshes), and gaming (D4 dice). The simplest polyhedron with 4 faces, 6 edges, 4 vertices.

How: The calculator uses the edge length a to derive all properties. Volume comes from the pyramid formula; surface area from four equilateral triangles.

Volume = a³√2/12 — derived from pyramid formula4 equilateral faces, 6 equal edges
Sources:Wolfram MathWorld — TetrahedronKhan Academy — Geometry

Run the calculator when you are ready.

Calculate Tetrahedron PropertiesEnter edge length to get all measurements
🔺
3D GEOMETRYPlatonic Solid

Tetrahedron — Simplest Polyhedron

4 equilateral faces, 6 equal edges. V = a³√2/12, A = √3·a².

🔺 Sample Examples — Click to Load

Regular Tetrahedron

tetrahedron_volume.sh
CALCULATED
$ calc --edge=5 --unit=cm
Volume
14.7314 cm³
Surface Area
43.3013 cm²
Height
4.0825 cm
Inradius
1.0206 cm
Circumradius
3.0619 cm
Share:
Tetrahedron Properties
V = 14.73 cm³
A = 43.30 cm² | h = 4.08
numbervibe.com/calculators/mathematics/3d-geometry/tetrahedron-volume-calculator

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

INPUT
Edge length a
5 cm
VOLUME
Volume V
14.7314 cm³
V = a^{3}√2/12
SURFACE
Surface Area A
43.3013 cm²
A = √3 cdot a^{2}
DIMENSIONS
Height h
4.0825 cm
h = a√(2/3)
Inradius r
1.0206 cm
r = a/(2√6)
Circumradius R
3.0619 cm
R = a√6/4

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

⚛️

Methane (CH₄) has tetrahedral molecular geometry — carbon at center.

— Chemistry

💎

Diamond crystal structure is based on tetrahedral carbon bonding.

— Crystallography

🎲

D4 dice are tetrahedral — 4 faces, each with 1/4 probability.

— Gaming

📐

Volume scales with a³ — doubling edge gives 8× volume.

— Geometry

📋 Key Takeaways

  • • Regular tetrahedron: 4 equilateral faces, 6 equal edges, 4 vertices
  • • Volume V=fraca3sqrt212V = \\frac{a^3\\sqrt{2}}{12} | Surface area A=sqrt3cdota2A = \\sqrt{3} \\cdot a^2
  • • One of 5 Platonic solids; simplest polyhedron
  • • Height h=asqrt2/3h = a\\sqrt{2/3}, inradius r=a/(2sqrt6)r = a/(2\\sqrt{6}), circumradius R=asqrt6/4R = a\\sqrt{6}/4

💡 Did You Know?

⚛️Methane (CH₄) has tetrahedral molecular geometry — carbon at center
💎Diamond crystal structure is based on tetrahedral carbon bonding
🎲D4 dice are tetrahedral — 4 faces, each with 1/4 probability
🏛️Plato associated the tetrahedron with fire in his cosmology
🔬Tetrahedral meshes are fundamental in 3D modeling and FEM
📐Volume scales with a³ — doubling edge gives 8× volume
🌐Many viruses have icosahedral shells built from tetrahedral subunits

📖 How It Works

Volume

V=fraca3sqrt212V = \\frac{a^3\\sqrt{2}}{12} — derived from pyramid formula V = (1/3)×base×height with equilateral base.

Surface Area

A=sqrt3cdota2A = \\sqrt{3} \\cdot a^2 — four equilateral triangles, each with area fracsqrt34a2\\frac{\\sqrt{3}}{4}a^2.

Inradius & Circumradius

Inradius: inscribed sphere touches faces. Circumradius: circumscribed sphere passes through vertices.

🎯 Expert Tips

Unit Consistency

Edge in cm → volume in cm³, area in cm².

Scaling

Volume ∝ a³. Double edge → 8× volume.

Platonic Solid

Only tetrahedron with triangular faces. All edges equal.

Irregular Tetrahedra

For irregular, use scalar triple product with vertex coordinates.

⚖️ Comparison

ShapeVolume Formula
Tetrahedrona³√2/12
Cubea³
Sphere4πr³/3

❓ FAQ

Tetrahedron vs pyramid?

Tetrahedron is a pyramid with triangular base — 4 triangular faces total.

How does edge length affect volume?

Volume ∝ a³. Double edge → 8× volume.

Irregular tetrahedron?

Use scalar triple product of edge vectors. This calculator is for regular only.

SA to volume ratio?

A/V = 12√3/(a√2). Smaller tetrahedra have higher ratio.

Where are tetrahedra in nature?

CH₄, diamond, many viruses, crystals, rock formations.

What is the dihedral angle?

~70.53° (arccos(1/3)) between any two faces.

📊 Stats

4
Faces
6
Edges
4
Vertices
5
Platonic Solids

⚠️ Disclaimer: Regular tetrahedron only. All edges equal. Use consistent units.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Ellipsoid Volume Calculator

Ellipsoid Volume Calculator - Calculate the volume and related properties of an ellipsoid with step-by-step solutions

Mathematics

Pyramid Volume Calculator

Pyramid Volume Calculator - Calculate the volume and related properties of a pyramid with step-by-step solutions

Mathematics

Torus Volume Calculator

Torus Volume Calculator - Calculate the volume and related properties of a torus with step-by-step solutions

Mathematics

Cube Calculator

Cube Calculator - Calculate volume, surface area, and other properties of a cube with step-by-step solutions

Mathematics

Sphere Calculator

Sphere Calculator - Calculate volume, surface area, and other properties of a sphere with step-by-step solutions

Mathematics

Cylinder Calculator

Cylinder Calculator - Calculate volume, surface area, and other properties of a right cylinder with step-by-step solutions

Mathematics