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

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

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

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.

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