# Volume & Surface Area of a Tetrahedron

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• 0:00 A Tetrahedron
• 0:44 Volume
• 1:59 Surface Area
• 2:45 Example
• 3:42 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Did you know that all you need to know to figure out the volume and surface area of a tetrahedron is the length of one of its sides? In this lesson, you'll learn what these two formulas are.

## A Tetrahedron

By definition, a tetrahedron is a solid with four equal equilateral triangles. It's a special solid, being one of the Platonic solids. A Platonic solid is a special three-dimensional solid whose faces are all the same and the same number of edges meet at each vertex. The tetrahedron has four faces, six edges, and four vertices. Three edges meet at each vertex.

Because the tetrahedron is a Platonic solid, there are formulas you can use to find its volume and surface area. These formulas actually make your job quite easy. All you need is one measurement, and you'll be able to find your answers.

## Volume

And, what is that one measurement? Well, because the tetrahedron is made up of four equilateral triangles, all its edges are the same. Remember, an equilateral triangle is a triangle with three congruent, or equal, sides. So you actually only have one value, and the same measurement for each and every edge of the tetrahedron. And, this is the only measurement you need to find your volume and surface area.

To find the volume of a tetrahedron, you'll use this formula:

The a stands for the length of one of the edges of the tetrahedron.

All you need to find the volume is the value for a. Once you have this number, then you can go ahead and plug it in and find your answer.

For example, if the tetrahedron has an edge length of 3 inches, then your volume would be calculated like this:

We multiply all of this out and get the volume of this tetrahedron with an edge of 3 inches is 3.18 cubic inches.

## Surface Area

For surface area, your formula is this one:

The a here also stands for the length of one of the edges of the tetrahedron. Using 2 inches for the edge of this tetrahedron, the surface area calculation looks like this:

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