Find The Volume Of The Tetrahedron Calculator

Easily calculate the volume of a tetrahedron using the coordinates of its four vertices. Input the x, y, and z coordinates for each vertex below to find the volume.

Calculate Volume of a Tetrahedron

Enter the coordinates of the four vertices (A, B, C, D) of the tetrahedron:

Input Coordinates and Derived Vectors

Vertex/Vector	X	Y	Z
A	0	0	0
B	1	0	0
C	0	1	0
D	0	0	1
Vector AB	1	0	0
Vector AC	0	1	0
Vector AD	0	0	1

Table showing the input coordinates and the calculated vectors AB, AC, and AD.

What is the Volume of a Tetrahedron?

The Volume of a Tetrahedron is a measure of the three-dimensional space enclosed by a tetrahedron. A tetrahedron is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. It is the simplest of all ordinary convex polyhedra and the only one with fewer than 5 faces.

Knowing the volume of a tetrahedron is important in various fields such as geometry, physics, engineering, and computer graphics. For example, it can be used to determine the amount of material in a pyramid-like structure or in finite element analysis.

This Volume of a Tetrahedron Calculator helps you find this volume easily if you know the coordinates of its four vertices.

Common misconceptions include thinking that all tetrahedrons are regular (with equilateral triangle faces), which is not true. Our calculator works for any tetrahedron, regular or irregular, defined by its vertices.

Volume of a Tetrahedron Formula and Mathematical Explanation

If you have the coordinates of the four vertices of a tetrahedron, say A=(x_A, y_A, z_A), B=(x_B, y_B, z_B), C=(x_C, y_C, z_C), and D=(x_D, y_D, z_D), you can find the volume using the scalar triple product.

1. Form three vectors from one vertex (say A) to the other three vertices:

• AB = (x_B – x_A, y_B – y_A, z_B – z_A)
• AC = (x_C – x_A, y_C – y_A, z_C – z_A)
• AD = (x_D – x_A, y_D – y_A, z_D – z_A)

2. Calculate the cross product of two of these vectors, for example, AC x AD.

3. Calculate the dot product of the remaining vector (AB) with the result of the cross product: AB · (AC x AD). This is the scalar triple product.

4. The volume of the tetrahedron is one-sixth of the absolute value of this scalar triple product:

Volume (V) = (1/6) * |AB · (AC x AD)|

Alternatively, if AB = (a₁, a₂, a₃), AC = (b₁, b₂, b₃), and AD = (c₁, c₂, c₃), the volume is (1/6) of the absolute value of the determinant:

V = (1/6) * | det

Variables Table

Variable	Meaning	Unit	Typical Range
xA, yA, zA	Coordinates of vertex A	Length units	Any real number
xB, yB, zB	Coordinates of vertex B	Length units	Any real number
xC, yC, zC	Coordinates of vertex C	Length units	Any real number
xD, yD, zD	Coordinates of vertex D	Length units	Any real number
V	Volume of the Tetrahedron	Cubic length units	≥ 0

Practical Examples (Real-World Use Cases)

Let's look at how to find the Volume of a Tetrahedron with some examples.

Example 1: A Simple Tetrahedron

Suppose the vertices are A=(0,0,0), B=(2,0,0), C=(0,3,0), and D=(0,0,4).

• AB = (2, 0, 0)
• AC = (0, 3, 0)
• AD = (0, 0, 4)
• AC x AD = (12, 0, 0)
• AB · (AC x AD) = 2*12 + 0*0 + 0*0 = 24
• Volume = (1/6) * |24| = 4 cubic units.

Example 2: An Irregular Tetrahedron

Vertices: A=(1,1,1), B=(2,3,1), C=(1,4,2), D=(3,2,5).

• AB = (1, 2, 0)
• AC = (0, 3, 1)
• AD = (2, 1, 4)
• AC x AD = (3*4 – 1*1, 1*2 – 0*4, 0*1 – 3*2) = (11, 2, -6)
• AB · (AC x AD) = 1*11 + 2*2 + 0*(-6) = 11 + 4 + 0 = 15
• Volume = (1/6) * |15| = 2.5 cubic units.

Our Volume of a Tetrahedron Calculator handles these calculations automatically.

How to Use This Volume of a Tetrahedron Calculator

1. Enter Vertex Coordinates: Input the x, y, and z coordinates for each of the four vertices (A, B, C, D) of the tetrahedron into the respective fields.
2. Calculate: The calculator will automatically update the volume and intermediate results as you type. You can also click the "Calculate Volume" button.
3. View Results: The primary result is the Volume of the Tetrahedron, displayed prominently. Intermediate results like the vectors AB, AC, AD, their cross product, and the scalar triple product are also shown.
4. Table and Chart: The table below the calculator summarizes the input coordinates and the derived vectors. The chart visualizes the magnitudes of the vectors AB, AC, and AD.
5. Reset: Click "Reset" to clear the fields and start with default values.
6. Copy Results: Click "Copy Results" to copy the volume and intermediate values to your clipboard.

The results give you the volume in cubic units corresponding to the units of your input coordinates. If the four vertices are coplanar, the volume will be zero.

Key Factors That Affect Volume of a Tetrahedron Results

• Vertex Coordinates: The most direct factor. Changing any coordinate of any vertex will likely change the volume. The relative positions of the vertices determine the lengths and orientations of the edges.
• Edge Lengths: While not direct inputs here, the distances between vertices define the edge lengths, which in turn define the volume.
• Angles Between Edges: The angles between the edges emanating from a vertex influence the area of the base and the height, thus the volume.
• Coplanarity of Vertices: If all four vertices lie on the same plane, the tetrahedron is "flat" and has zero volume.
• Choice of Origin Vertex (for vectors): While the final volume is independent of which vertex is chosen as the origin for the three vectors (e.g., A in our case), the intermediate vector components will differ.
• Units of Coordinates: The volume will be in cubic units of whatever unit was used for the coordinates (e.g., cubic meters if coordinates were in meters).

Frequently Asked Questions (FAQ)

What is a tetrahedron?

A tetrahedron is a solid with four triangular faces, four vertices (corners), and six edges. It's the simplest polyhedron.

What is a regular tetrahedron?

A regular tetrahedron has four equilateral triangles as its faces. All its edges are of equal length, and all faces are congruent.

How is the volume of a tetrahedron calculated using vertices?

The volume is calculated as 1/6 of the absolute value of the scalar triple product of three vectors formed by taking one vertex as the origin and connecting it to the other three vertices. See the tetrahedron volume formula section above.

Can the volume of a tetrahedron be zero?

Yes, if the four vertices lie on the same plane (are coplanar), the volume is zero.

What units will the volume be in?

The volume will be in cubic units of the length unit used for the coordinates (e.g., cm³ if coordinates are in cm).

Does the order of vertices matter when forming vectors?

The order in which you take the cross product and then the dot product can change the sign of the scalar triple product, but since we take the absolute value for the volume, the final volume remains the same.

Can I use this calculator for any type of tetrahedron?

Yes, this Volume of a Tetrahedron Calculator works for any tetrahedron, regular or irregular, as long as you provide the coordinates of its four vertices.

What is the scalar triple product?

The scalar triple product of three vectors a, b, and c is given by a · (b x c). Geometrically, its absolute value represents the volume of the parallelepiped formed by the three vectors.