Find The Volume Of The Tetrahedron With Vertices Calculator

Calculate Tetrahedron Volume

Enter the coordinates of the four vertices A, B, C, and D.

Vertex A:

Vertex B:

Vertex C:

Vertex D:

Vector	x-component	y-component	z-component	Magnitude
AB	0	0	0	0.00
AC	0	0	0	0.00
AD	0	0	0	0.00

Components and magnitudes of vectors AB, AC, and AD.

What is a Volume of Tetrahedron with Vertices Calculator?

A volume of tetrahedron with vertices calculator is a tool used to determine the volume of a tetrahedron when the Cartesian coordinates (x, y, z) of its four vertices are known. 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.

This calculator is particularly useful for students, engineers, mathematicians, and physicists who need to find the volume of such a 3D shape defined by four points in space. Instead of manually calculating the scalar triple product or determinant, the volume of tetrahedron with vertices calculator automates the process, providing quick and accurate results.

Common misconceptions include thinking that any four points form a non-degenerate tetrahedron (they could be coplanar, forming a flat shape with zero volume) or that the order of vertices doesn't matter (while the volume magnitude remains the same, the sign of the scalar triple product changes with order, but we take the absolute value for volume).

Volume of Tetrahedron with Vertices Calculator Formula and Mathematical Explanation

The volume of a tetrahedron with vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3), and D(x4, y4, z4) can be found using the scalar triple product of three vectors originating from one vertex, say A, to the other three vertices (AB, AC, AD).

1. First, define the vectors from vertex A to vertices B, C, and D:

• AB = (x2 – x1, y2 – y1, z2 – z1)
• AC = (x3 – x1, y3 – y1, z3 – z1)
• AD = (x4 – x1, y4 – y1, z4 – z1)

2. The volume of the parallelepiped formed by these three vectors is given by the absolute value of their scalar triple product: | AB · (AC x AD) |. The scalar triple product can also be calculated as the determinant of the matrix whose rows (or columns) are the components of these vectors:

Scalar Triple Product = det | (x2-x1) (y2-y1) (z2-z1) | | (x3-x1) (y3-y1) (z3-z1) | | (x4-x1) (y4-y1) (z4-z1) |

3. The volume of the tetrahedron is 1/6th of the volume of this parallelepiped:

Volume = (1/6) * | (x2-x1)((y3-y1)(z4-z1) – (y4-y1)(z3-z1)) – (x3-x1)((y2-y1)(z4-z1) – (y4-y1)(z2-z1)) + (x4-x1)((y2-y1)(z3-z1) – (y3-y1)(z2-z1)) |

Or more compactly, Volume = | AB · (AC x AD) | / 6.

Variables Table

Variable	Meaning	Unit	Typical Range
x1, y1, z1	Coordinates of vertex A	Length units	Real numbers
x2, y2, z2	Coordinates of vertex B	Length units	Real numbers
x3, y3, z3	Coordinates of vertex C	Length units	Real numbers
x4, y4, z4	Coordinates of vertex D	Length units	Real numbers
Volume	Volume of the tetrahedron	Cubic length units	Non-negative real numbers

Variables used in the volume of tetrahedron calculation.

Practical Examples (Real-World Use Cases)

Example 1: Simple Tetrahedron

Let's find the volume of a tetrahedron with vertices A=(0,0,0), B=(3,0,0), C=(0,4,0), and D=(1,1,5).

• A = (0, 0, 0)
• B = (3, 0, 0)
• C = (0, 4, 0)
• D = (1, 1, 5)

Vectors: AB = (3,0,0), AC = (0,4,0), AD = (1,1,5)

Scalar triple product = 3 * (4*5 – 0*1) – 0 + 0 = 3 * 20 = 60

Volume = |60| / 6 = 10 cubic units. The volume of tetrahedron with vertices calculator would confirm this.

Example 2: Another Tetrahedron

Consider vertices A=(1,1,1), B=(2,3,4), C=(4,2,3), D=(3,4,2).

• A = (1, 1, 1)
• B = (2, 3, 4)
• C = (4, 2, 3)
• D = (3, 4, 2)

Vectors: AB = (1,2,3), AC = (3,1,2), AD = (2,3,1)

Scalar triple product = 1(1*1 – 2*3) – 2(3*1 – 2*2) + 3(3*3 – 1*2) = 1(-5) – 2(-1) + 3(7) = -5 + 2 + 21 = 18

Volume = |18| / 6 = 3 cubic units. Using a volume of tetrahedron with vertices calculator is much faster.

How to Use This Volume of Tetrahedron with Vertices Calculator

1. Enter Vertex Coordinates: Input the x, y, and z coordinates for each of the four vertices A, B, C, and D into the respective input fields.
2. Calculate: The calculator automatically updates 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 components of vectors AB, AC, AD, and the scalar triple product are also shown.
4. Vector Table & Chart: The table below the results shows the components and magnitudes of vectors AB, AC, and AD. The chart visually represents these magnitudes.
5. Reset: Click "Reset" to clear the fields to default values.
6. Copy Results: Click "Copy Results" to copy the volume and intermediate values to your clipboard.

The volume of tetrahedron with vertices calculator provides immediate feedback, allowing for quick analysis.

Key Factors That Affect Volume of Tetrahedron Results

• Vertex Coordinates: The most direct factor. Changing any coordinate of any vertex will alter the vectors AB, AC, AD, and thus the volume.
• Collinearity/Coplanarity: If three vertices are collinear (on the same line), or if all four vertices are coplanar (on the same plane), the volume will be zero, indicating a degenerate tetrahedron.
• Units of Coordinates: The volume will be in cubic units of the length unit used for the coordinates (e.g., if coordinates are in cm, volume is in cm³).
• Relative Positions: The spatial arrangement of the vertices relative to each other determines the lengths of the edges and the angles between them, which collectively define the volume.
• Orientation: While the volume (a scalar) doesn't have orientation, the sign of the scalar triple product depends on the order of vectors (right-hand vs. left-hand system), but we take the absolute value for volume.
• Accuracy of Input: Small errors in coordinate input can lead to different volume results, especially for nearly degenerate tetrahedrons. Using our precise volume of tetrahedron with vertices calculator helps minimize manual calculation errors.

Frequently Asked Questions (FAQ)

What if the four vertices lie on the same plane?

If the four vertices are coplanar, the volume of the tetrahedron will be zero. The volume of tetrahedron with vertices calculator will show 0.

Does the order of vertices matter when inputting?

For the volume calculation (which takes the absolute value), the order in which you define A, B, C, D doesn't change the final volume magnitude. However, the sign of the scalar triple product will change.

What units will the volume be in?

The volume will be in cubic units corresponding to the units used for the coordinates. If you input coordinates in meters, the volume is in cubic meters.

Can I use this calculator for a regular tetrahedron?

Yes, if you know the coordinates of the vertices of a regular tetrahedron, you can input them to find its volume. For a regular tetrahedron, all edge lengths are equal.

What is the scalar triple product?

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

How is the volume of a tetrahedron related to a pyramid?

A tetrahedron is a type of pyramid with a triangular base. Its volume is (1/3) * (base area) * height, which is consistent with the (1/6) * scalar triple product formula.

Can the volume be negative?

Volume itself is a non-negative quantity. The scalar triple product can be negative, but we take its absolute value to find the volume using the volume of tetrahedron with vertices calculator.

What if I only know edge lengths?

If you only know the edge lengths, you would use a different formula (like the Cayley-Menger determinant) or first determine the coordinates of the vertices based on the edge lengths, then use this volume of tetrahedron with vertices calculator.