Dodecahedron: Volume & Surface Area Formulas

Instructor: Mark Boster
'Dodecahedron' may be a difficult word to spell, but this 12-faced figure is easy to grasp. Let's explore the formulas for surface area and volume of the dodecahedron.

What is a Dodecahedron?

You may have heard some compound words like 'doghouse,' 'snowflake,' and 'notebook.' But did you know there's a special compound word in math for a shape that has 12 sides? The word 'dodecahedron' comes from the Greek words dodeca ('twelve') and hedron ('faces'). So, a dodecahedron is a 3-dimensional figure made up of 12 faces, or flat sides. All of the faces are pentagons of the same size.

Surface Area of a Dodecahedron

Now, let's learn how to find the surface area of a dodecahedron. Surface area is the amount of space that all the faces of the shape take up.

Formula for the surface area (A) of a dodecahedron
null

The formula may look complicated, but when you work through it step-by-step, it's not so difficult. Say that we have a dodecahedron where the length of one edge is 6 inches. (Remember that an edge is one edge of one of the pentagonal faces.)

Dodecahedron with edges measuring 6 inches
null

Let's go through the formula step-by-step:

  • Begin by finding the square root of 5, which gives you 2.236.
  • Multiply your answer by 10, which gives you 22.36.
  • Add 25 to get 47.36.
  • Find the square root of this sum, which is 6.88186.
  • Multiply the above answer by 3, which is 20.64558.
  • Now, multiply the above number by e squared, where e is the length of an edge. In our example, the edge is 6 inches, so e squared is 36. So, 20.64 x 36 = 743.24.

The surface area of our dodecahedron is 743.24 square inches.

Volume of a Dodecahedron

Perhaps you want to find the volume of the same dodecahedron, which is the amount of space that the shape takes up. Let's take a look at the formula:

Formula for the volume (V) of a dodecahedron
null

Again, it may look complicated, but it's easier when we solve the formula step-by-step. Let's find the volume of our dodecahedron with an edge measuring 6 inches:

  • First, find the square root of 5, which is 2.236.
  • Multiply that by 7, which gives you 15.652.
  • Add 15 to get 30.652.
  • Divide the above sum by 4, which gives you 7.663.
  • Now, multiply the above answer by e cubed. (Remember that e is 6, so e cubed is 216.) So, 7.663 x 216 = 1655.208.

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