Understand the regular tetrahedron

A regular tetrahedron has four congruent equilateral triangular faces, six equal edges and four vertices. Three faces meet at each vertex. An arbitrary tetrahedron is not determined by one edge and is outside these calculators.

Properties that make the calculations work

Its height is a√(2/3), its surface area is √3a², and its volume is a³/(6√2). The altitude reaches the centre of the opposite equilateral face.

Where the model is useful

Use edge, height, surface area or volume to size an ideal regular tetrahedron. The edge is the shared side of its equilateral faces; the solid’s height is not the altitude within one face.

Measure the right thing

These formulas require all six edges to be equal. A face altitude and the solid’s vertical height are different.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k² and any volume by k³. This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area and cm³ for volume; the unit selector relabels rather than converts entered numbers.

Continue learning

Wolfram MathWorld: Regular tetrahedron provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.