How to Calculate the Volume of a Regular Dodecahedron

Step-by-Step Guide

1. 1

   Step 1: Remember or write down the formula 15+754(a3){\displaystyle {\frac {15+7{\sqrt {5}}}{4}}(a^{3})}.

   This formula will be used to calculate the volume of a regular dodecahedron. ,  There is a picture for you to give the idea on what a{\displaystyle a} stands for.
   
   Example:
   If a problem says that a=3{\displaystyle a=3}, you can plug the value in as V=15+754(a3)=15+754(33){\displaystyle V={\frac {15+7{\sqrt {5}}}{4}}(a^{3})={\frac {15+7{\sqrt {5}}}{4}}(3^{3})}. ,  Note that 5{\displaystyle {\sqrt {5}}} is similar to
   2.24.
   
   Example: 15+754(33)=15+15.684(33){\displaystyle {\frac {15+7{\sqrt {5}}}{4}}(3^{3})={\frac {15+15.68}{4}}(3^{3})}. , Example: 15+15.684(33)=30.684(33){\displaystyle {\frac {15+15.68}{4}}(3^{3})={\frac {30.68}{4}}(3^{3})}. , Example:
   30.684(33)=7.67(33){\displaystyle {\frac {30.68}{4}}(3^{3})=7.67(3^{3})} , Example:
   7.67(33)=7.67(27){\displaystyle
   7.67(3^{3})=7.67(27)} , Example:
   V=7.67(27)=207.69{\displaystyle V=7.67(27)=207.69}.
   
   Therefore, the volume of this dodecahedron is about
   207.69
2. 2

   Step 2: Find the value of the side length and replace a{\displaystyle a}.

3. 3

   Step 3: Using the order of operations (PEMDAS)

4. 4

   Step 4: multiply 7{\displaystyle 7} by 5{\displaystyle {\sqrt {5}}}.

5. 5

   Step 5: Add 15 and 15.68 together.

6. 6

   Step 6: Divide the sum by 4.

7. 7

   Step 7: Cube a

8. 8

   Step 8: Finally

9. 9

   Step 9: multiply 7.67 by the product.

Detailed Guide

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