Tetrahedron Volume Calculator

A tetrahedron is the simplest three-dimensional shape bounded by faces. It consists of four triangles, four vertices and six edges. If all the edges have the same length, it's called a regular tetrahedron whose four sides are identical equilateral triangles. This calculator deals with this case and allows calculations in any direction. Even if only one measurement is available, whether it be edge length, height, surface area or volume, it can be entered to find the remaining three values. By entering a number and selecting the unit, the volume will be displayed as the main result.

What is a tetrahedron?

The tetrahedron is one of the five platonic solids and is next to cube, octahedron, dodecahedron and icosahedron. It consists of four equilateral triangles, has six edges and four corners, with three surfaces meeting at each corner. Since the entire shape is determined only by the length of a single edge, all other sizes such as height, surface area, volume and the size of inner and outer spheres can be derived from this one value.

How to use this calculator:

Enter any one measurement into its corresponding input field and select the unit. After entering a value, the remaining three fields will be automatically filled out, with volume being displayed as the main result. It is not necessary to tell the calculator which measurement you are working with. If you want to recalculate using a different value, simply erase the previously entered values and enter new ones. Each field has an independent unit menu, so for example, edge length can be entered in inches and volume can be displayed in cubic centimeters if desired.

Formula for volume of a regular tetrahedron:

All calculations are based on the length of the edges, denoted as 'a'. The volume of a regular tetrahedron is obtained by multiplying the cube of the edge length with the square root of two and dividing it by twelve.

The whole surface consists of four equilateral triangles and can be simplified by multiplying the square root of three with the square of the length of an edge. The height from the ground to the top is calculated using the square root of six.

We check the validity using a tetrahedron with an edge length of 10 cm. The volume is approximately 117.85 cubic centimeters (calculated by multiplying the cube of the edge length by the square root of two and then dividing it by twelve). The surface area is approximately 173.21 square centimeters (calculated by multiplying the square root of three with one hundred). The height is approximately 8.16 cm. The square roots occur because the apex does not lie directly above a vertex of the base, but rather over the centroid of the base. By drawing a perpendicular and applying Pythagoras' theorem, we obtain the square roots of two, three and six as results.

Calculation based on volume:

It is not necessary to know the length of the edges in advance. Since volume is proportional to the cube of edge length, you can use the cubic root to find the edge length from the desired volume.

This is the process a calculator tool goes through when you input a volume. It calculates the length of the edges and then determines the corresponding height and surface area. The same procedure can be used to determine the edge length from the surface area. This is because 'a' is equal to the square root of the value obtained by dividing the surface area (A) by the square root of three.

List of all measurements for a regular tetrahedron.

Since each measurement is related to the edge length, the remaining values can be calculated when any one of them is entered. The following table shows the formulas used for calculation.

Examples of calculations:

This table shows the volume for typical side lengths and the results are rounded to two decimal places. Note that the volume increases very quickly as the side length gets larger since it is proportional to the cube of the side length.

The field "20 mm" refers to a standard tetrahedron cube (d4) for role-playing games. This tetrahedron cube with an edge length of 20 mm contains about 943 cubic millimeters of material, which is just under one cubic centimeter.

Spheres associated with tetrahedron.

A regular tetrahedron has three interlocking spheres, each of which is proportional to the length of its edges. The inscribed sphere lies inside the tetrahedron and touches every face. The circumscribed sphere passes through all four vertices. The midsphere touches each edge at its midpoint. From these formulas a simple fact can be deduced: the radius of the circumscribed sphere is exactly three times that of the inscribed sphere.

Angle and Dihedral Angle and Bond Angle of Methane

A regular tetrahedron has two important angles that are constant regardless of its size. The dihedral angle is the angle between two faces, measured along their common edge, and it is equal to the arccosine of one third, which is approximately 70.53 degrees. The tetrahedral angle is the angle from a vertex through the center to another vertex, and it is equal to the arccosine of minus one third, which is approximately 109.47 degrees. The second angle is the well-known bond angle of methane. This is the H-C-H bond angle in methane, and also represents the geometric configuration of all sp3-hybridized carbons. This is one reason why diamond is so hard. You can check these two angles by opening up the plane of symmetry.

Irregular tetrahedrons

All of the above formulas assume that all six edges are the same length. If it is a general triangular pyramid where each edge has a different length than the others, then the regular tetrahedron formulas do not apply, but the rules for volume of a pyramid still hold true. The volume of any tetrahedron is one-third the product of its base area and its perpendicular height from the base to the opposite vertex.

If you open the general tetrahedron plane and enter the base area and vertical height, then the calculator will give you the volume. For a fully irregular tetrahedron which is only defined by the length of its six edges, these six numbers can be converted into the volume using the Cayley–Menger determinant.

The tetrahedron's fields of application:

This shape is widely used in science and design. In chemistry tetrahedral molecular structures represent methane, silicate minerals, and sp3 carbon atoms with all bond angles of 109.47 degrees. In crystallography the diamond lattice that forms the basis for diamonds, silicon, and germanium consists of interconnected tetrahedrons. Engineers and architects use tetrahedral trusses and space frames which is a typical design by Buckminster Fuller. As a tetrahedron cannot be deformed without lengthening at least one of its edges it is the most stable simple frame construction in three-dimensional space. In board games four-sided dice (d4) are themselves tetrahedrons. The shape is also used in computer graphics where the volume of a mesh is usually calculated by subdividing the body into tetrahedrons and adding up their volumes.

Tips for accurate results:

Measure the sides carefully as their length will affect the result of the volume in the cube. Small errors in measuring the side length will lead to larger errors in the calculated volume. Be aware which height is meant. The height of a regular tetrahedron from its base to its apex is different than the face height. Do not confuse a tetrahedron with a square pyramid like the Great Pyramid of Giza. The latter has five faces instead of four. Use consistent units and choose your desired output unit so that the calculator will perform the conversion for you.