Tetrahedron volume calculator

What is a tetrahedron?

A tetrahedron is a three-dimensional polyhedron with four triangular faces, six edges, and four vertices. It is the simplest of all ordinary convex polyhedra. A regular tetrahedron has all edges of equal length, and all faces are equilateral triangles. In contrast, an irregular tetrahedron has edges of varying lengths and faces that may be scalene or isosceles triangles. The tetrahedron is one of the five Platonic solids and has been studied since antiquity, with references dating back to ancient Greek mathematicians like Euclid.

Formula for calculating the volume of a tetrahedron

Volume using base area and height

For any tetrahedron, if the area of the base $A$ and the height $h$ (perpendicular distance from the base to the opposite vertex) are known, the volume is:

This formula is analogous to the volume of a pyramid and applies universally to all tetrahedrons, whether regular or irregular.

Regular tetrahedron volume formula

For a regular tetrahedron with edge length $a$, the volume $V$ is calculated using: $V = \frac{\sqrt{2}}{12} \times a^3$ or it can also be written in the form:

This formula derives from the relationship between the edge length and the height of the tetrahedron, leveraging geometric symmetry.

Irregular tetrahedron volume formula

For an irregular tetrahedron defined by vertices $A, B, C, D$, the volume can be calculated using the scalar triple product of vectors originating from one vertex. If vectors $\vec{AB}$, $\vec{AC}$, and $\vec{AD}$ are known, the volume is:

This method works for any tetrahedron, regardless of symmetry.

Examples of volume calculations

Example 1: Regular tetrahedron

Problem: Calculate the volume of a regular tetrahedron with an edge length of 5 cm.
Solution:
Substitute $a = 5$ into the formula:

Example 2: Irregular tetrahedron

Problem: Find the volume of a tetrahedron with vertices at $A(0, 0, 0)$, $B(2, 0, 0)$, $C(0, 3, 0)$, and $D(0, 0, 4)$.
Solution:

1. Define vectors from vertex $A$: $$\vec{AB} = (2, 0, 0), \quad \vec{AC} = (0, 3, 0), \quad \vec{AD} = (0, 0, 4)$$
2. Compute the cross product $\vec{AC} \times \vec{AD}$: $$\vec{AC} \times \vec{AD} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{vmatrix} = (12, 0, 0)$$
3. Compute the dot product $\vec{AB} \cdot (\vec{AC} \times \vec{AD})$: $$(2, 0, 0) \cdot (12, 0, 0) = 2 \times 12 + 0 + 0 = 24$$
4. Calculate the volume: $$V = \frac{1}{6} \times |24| = 4 \, \text{units}^3$$

Example 3: Volume using base area and height

Problem: A tetrahedron has a triangular base with an area of 24 cm². The height from the base to the opposite vertex is 9 cm. What is its volume?
Solution:
Using the formula $V = \frac{1}{3} A h$:

Notes

1. For irregular tetrahedrons, ensure vectors are defined from the same vertex.
2. Units must be consistent (e.g., all edges in centimeters).
3. The regular tetrahedron’s volume formula is a special case of the general scalar triple product method.
4. The formula $V = \frac{1}{3} A h$ is particularly useful when the base shape is known but the tetrahedron is not regular.
5. Online calculators automate these computations, reducing manual errors.

Frequently Asked Questions

How does edge length affect the volume of a regular tetrahedron?

The volume of a regular tetrahedron is proportional to the cube of its edge length. For example, doubling the edge length increases the volume by $2^3 = 8$ times.

Can the volume of an irregular tetrahedron be zero?

Yes. If all four vertices lie on the same plane, the scalar triple product becomes zero, resulting in zero volume.

What is the difference between regular and irregular tetrahedrons?

A regular tetrahedron has all edges equal and equilateral triangular faces, while an irregular tetrahedron has edges of varying lengths and non-equilateral faces.

How to use the scalar triple product for volume calculation?

1. Choose one vertex as the origin.
2. Compute vectors from this vertex to the other three vertices.
3. Calculate the scalar triple product of these vectors.
4. Divide the absolute result by 6 to get the volume.

Why is the denominator $6\sqrt{2}$ in the regular tetrahedron formula?

The term $\sqrt{2}$ arises from the Pythagorean relationship in the tetrahedron’s geometry, and the denominator 6 scales the result to match the unit volume.