Tetrahedron: Definition & Formula

Tetrahedron: Definition & Formula
Coming up next: Ratios and Proportions: Definition and Examples

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 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
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed

Recommended Lessons and Courses for You

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

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create An Account
Support