Tetrahedron Volume Calculator - Volume from Edge or Base

The tetrahedron volume calculator finds the three-dimensional space inside any tetrahedron from either one edge length of a regular tetrahedron or from a triangular base area and a perpendicular height for a general tetrahedron. Use it when a geometry problem, a chemistry tetrahedral void estimate, a tetrahedral mesh volume check, or a textbook exercise asks for the volume.

• • Classroom geometry: Check V = a^3 / (6 sqrt(2)) for regular tetrahedra and V = (1/3) B h for any tetrahedron, with each step exposed in the result.
• • Chemistry and crystal geometry: Estimate the volume of a tetrahedral void or unit cell from a single edge length or from a known base area and height.
• • 3D modeling and mesh checks: Cross-check the volume reported by a CAD or 3D modeler against a hand calculation built from the same edge or base measurements.
• • Quick sanity checks: Compare a hand-computed tetrahedron volume against the calculator before using the value in a larger formula.

A tetrahedron is the simplest 3D solid with flat faces: four triangular faces, four vertices, and six edges. A regular tetrahedron has all six edges equal and four equilateral triangle faces. A general tetrahedron keeps the four-triangle structure but allows edges and faces to vary, so the calculator treats it as a triangular pyramid.

The result is a volume, not an area. Keep the input lengths in the same unit. If you enter meters, the volume is cubic meters; if you enter inches, the volume is cubic inches.

When the tetrahedron has a non-equilateral triangular base, the Triangle Calculator helps derive the irregular base area before switching to the base-area input mode on this page.

The tetrahedron volume calculator switches between two formula paths based on the chosen input mode. In regular mode it cubes the edge length and divides by six times the square root of two. In base-area mode it uses the same V = (1/3) B h rule that governs every pyramid.

• a: edge length of the regular tetrahedron (all six edges equal)
• B: triangular base area for the general tetrahedron
• h: perpendicular height from the base plane to the opposite vertex
• sqrt(2): fixed constant in the regular tetrahedron volume factor 1 / (6 sqrt(2))

The general pyramid volume formula is V = (1/3) B h. For regular mode, B is the equilateral triangle area (sqrt(3)/4) a^2 and h is a * sqrt(2/3), which together simplify to V = a^3 / (6 sqrt(2)).

The calculator keeps full precision internally and rounds each displayed output to four decimal places so the values are easy to compare against a hand calculation. The base area and perpendicular height outputs help audit the intermediate steps.

According to Wolfram MathWorld, the volume of a regular tetrahedron is a^3 / (6 sqrt(2)) and the perpendicular height is a * sqrt(2/3).

For a pyramid with a square base and the same vertical height, the Square Pyramid Volume Calculator applies V = (1/3) B h with a different base area, so the two pages are useful side-by-side references.

These four ideas decide whether the tetrahedron volume formula matches the solid you are measuring.

The most common mistake is using slant height where the tetrahedron volume calculator asks for perpendicular height. If you only have the slant height along one of the triangular faces, first convert it to perpendicular height using the base geometry as a right-triangle leg.

If the tetrahedron is not regular, switching to base-area mode keeps the same V = (1/3) B h logic but lets you enter the irregular base area and perpendicular height directly, which avoids having to back out a single effective edge length from a non-equilateral triangle.

For a regular pyramid with a six-sided base, the Volume Hexagonal Pyramid Calculator uses the same V = (1/3) B h rule with a different base area and a different edge-to-height relationship.

Pick the input mode that matches the data you have, enter it with consistent length units, and read the volume along with the supporting base area and perpendicular height.

1. 1 Choose the input mode: Select 'Regular tetrahedron' for one edge length, or 'Base area and height' when the triangular base area and perpendicular height are known.
2. 2 Enter the regular edge length: Type a single edge length a for the regular tetrahedron. The calculator derives the equilateral base area and the perpendicular height for you.
3. 3 Or enter base area and height: For a general tetrahedron, type the triangular base area in matching square units and the perpendicular height in the same length unit.
4. 4 Read the volume: Use the volume as the cubic capacity inside the tetrahedron, paired with the units implied by the input length unit.
5. 5 Check the base area and height: Use the base area to confirm the triangular footprint and the perpendicular height to confirm the vertical distance.

Suppose a chemistry problem asks for the volume of a regular tetrahedral void with edge length 4 angstroms. Set the mode to 'Regular', enter 4, and the calculator returns 3.1748 cubic angstroms, an equilateral base area of 6.9282 square angstroms, and a perpendicular height of 3.2660 angstroms. The cubic-angstrom value is what flows into the next packing-fraction step.

After calculating cubic inches, cubic centimeters, or cubic angstroms, the Volume Converter can convert the finished volume into a different cubic unit.

Switching between regular and base-area inputs in one calculator keeps the formula choice out of the user's hands.

• • Two input paths, one result: Use a single edge length for a regular tetrahedron or base area and height for a general tetrahedron without switching to a different tool.
• • Audited intermediate steps: The triangular base area and perpendicular height outputs match the steps a teacher or worksheet would show, so the final volume is easy to verify.
• • Two formulas, one panel: The page handles V = a^3 / (6 sqrt(2)) and V = (1/3) B h in one calculator, so the user does not have to know which formula applies before opening the page.
• • Unit consistency: The volume is returned in cubic units that match the input length unit, with no silent conversions between metric and imperial.
• • Decimal support: Decimal edge lengths, base areas, and heights work for scaled drawings, models, and measured solids.

Because the calculator exposes the base area and the perpendicular height, you can decide whether to reuse those numbers. A surface area problem might need the equilateral face area; a packing-fraction problem might need the height-to-edge ratio.

The outputs also make errors easier to spot. If the base area is wrong, the volume will be wrong. If the base area looks too small, double-check whether you typed an edge length or a face height on the equilateral triangle.

For other solids such as prisms, cylinders, cones, and spheres, the Volume Calculator groups those shape-specific volume formulas in one place once the tetrahedron is done.

The formula is compact, but a few measurement choices affect whether the volume matches the solid.

• • This calculator does not solve for the perpendicular height from a slant height. If you only have slant height, first convert it to perpendicular height using the base geometry as a right-triangle leg.
• • It does not compute the volume directly from a list of six edge lengths. For that case, use the Cayley-Menger determinant on a separate page, then enter the resulting base area and height here.
• • The result is a geometric ideal. Real tetrahedral voids, meshes, and crystal cells include packing corrections and tolerance bands, so add an uncertainty margin before using the volume in a downstream simulation.

The regular tetrahedron volume factor 1 / (6 sqrt(2)) comes from combining the equilateral triangle area (sqrt(3)/4) a^2 with the perpendicular height a * sqrt(2/3), which is why the edge length is cubed in the final formula.

The base-area input mode exists for irregular tetrahedra, where the equilateral face area formula does not apply and a tilted face means the user should supply the triangular base area directly.

According to Wolfram MathWorld, the area of an equilateral triangle is (sqrt(3)/4) times the side length squared.

According to OpenStax, the volume of a pyramid equals one-third of the base area times the height of the pyramid.

If the base is round rather than triangular, the Cone Volume Calculator uses the same V = (1/3) B h rule with a circular base area instead of a triangular base area.