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Regular Tetrahedron — Definition, Formula & Examples

A regular tetrahedron is a three-dimensional solid made of exactly four faces, each of which is an equilateral triangle of the same size. It is the simplest of the five Platonic solids, with 4 vertices and 6 edges.

A regular tetrahedron is a convex polyhedron with four congruent equilateral triangular faces, four vertices, and six edges of equal length. Equivalently, it is the Platonic solid for which three equilateral triangles meet at every vertex.

Key Formula

V=a362A=3 a2V = \frac{a^3}{6\sqrt{2}} \qquad A = \sqrt{3}\,a^2
Where:
  • aa = Edge length of the regular tetrahedron
  • VV = Volume
  • AA = Total surface area

Worked Example

Problem: Find the surface area and volume of a regular tetrahedron with edge length 6 cm.
Surface Area: Use the surface area formula with a = 6.
A=3 (6)2=363≈62.35 cm2A = \sqrt{3}\,(6)^2 = 36\sqrt{3} \approx 62.35 \text{ cm}^2
Volume: Use the volume formula with a = 6.
V=6362=21662=362=182≈25.46 cm3V = \frac{6^3}{6\sqrt{2}} = \frac{216}{6\sqrt{2}} = \frac{36}{\sqrt{2}} = 18\sqrt{2} \approx 25.46 \text{ cm}^3
Answer: The surface area is 363≈62.3536\sqrt{3} \approx 62.35 cm² and the volume is 182≈25.4618\sqrt{2} \approx 25.46 cm³.

Why It Matters

Regular tetrahedra appear in chemistry as the shape of molecules like methane (CH₄) and in engineering for constructing rigid trusses. Understanding this solid also builds the foundation for studying all five Platonic solids and for verifying Euler's formula on simple polyhedra.

Common Mistakes

Mistake: Confusing a regular tetrahedron (4 equilateral triangular faces) with a triangular pyramid that has a non-equilateral base.
Correction: Every face of a regular tetrahedron must be the same equilateral triangle. If the base is a different triangle or the lateral faces are isosceles, the solid is not a regular tetrahedron.