# Tetrahedron: Definition & Formula

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• 0:01 Defining the Tetrahedron
• 0:38 Volume Formula in General
• 2:26 Volume of a Regular…
• 3:25 Surface Area
• 4:14 Lesson Summary

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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson, discover the solid called the tetrahedron and use the provided formulas to find its volume and surface area. Special formulas and examples for regular tetrahedra are included.

## Defining the Tetrahedron

A tetrahedron, or the plural tetrahedra, is simply a pyramid with a triangular base. So no, not like the ones in Egypt. It is a solid object with four triangular faces, three on the sides or lateral faces, and one on the bottom or the base, and four vertices or corners. If the faces are all congruent equilateral triangles, then the tetrahedron is called regular.

In fact, you may have seen and held in your hands a regular tetrahedron. Some games use 4-sided dice in the shape of regular tetrahedra.

## Volume Formula in General

Because the tetrahedron is a type of pyramid, its volume formula is the same as for all pyramids:

In this formula, B is the area of the base, and h is the height.

For example, a tetrahedron with a height of 10 inches and base triangle that has an area of 12 square inches, would have a volume of one third times 12 times 10. Volume equals 40 cubic inches.

The next example is a little more challenging. Suppose there is a tetrahedron with vertices located at the points A = (0, 0, 0), B = (5, 0, 0), C = (0, 6, 0), and D = (0, 0, 7) . What's the volume of tetrahedron ABCD?

It helps to remember a little bit of coordinate geometry in order to identify the lengths of segments AB, AC, and AD. Since these segments are parallel to the x, y, and z axes respectively, the lengths are easy to find: AB = 5, AC = 6, and AD = 7. Now move on to the volume computation.

First find the base area. The base in this case is a right triangle, base AC = 6, and height AB = 5 - not to be confused with the height of the tetrahedron itself! So, the base area is (1/2)(6)(5) = 15. This is B in the volume formula above. Next, the height of the solid is h = AD = 7, so the volume of the tetrahedron is:

V = (1/3)(15)(7) = 35

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