Finding the volume of a pyramid with a square base involves using a specific formula that relates the base area and the height of the pyramid. This guide will explain the necessary formula, show how to apply it, and provide an example calculation to help you understand the process.
Volume of a Pyramid with a Square Base Formula
To calculate the volume (\( V \)) of a pyramid with a square base, you can use the following formula:
\[ V = a^2 \cdot \dfrac{h}{3} \]
Where:
- \( a \) is the length of one side of the square base.
- \( h \) is the height of the pyramid.
Explanation of the Formula
- The term \( a^2 \) represents the area of the square base (since the area of a square is side length squared).
- The height (\( h \)) is the perpendicular distance from the base to the apex (top point) of the pyramid.
- The factor \(\dfrac{1}{3}\) accounts for the fact that the volume of a pyramid is one-third of the volume of a prism with the same base area and height.
Step-by-Step Calculation
Let's walk through an example to illustrate how to use this formula.
Example: Calculating the Volume of a Pyramid with a Square Base
1. Identify the given values:
- Side length of the base (\( a \)) = 6 units
- Height of the pyramid (\( h \)) = 9 units
2. Substitute the values into the volume formula:
\[ V = 6^2 \cdot \dfrac{9}{3} \]
3. Simplify the expression inside the formula:
\[ 6^2 = 36 \]
\[ \dfrac{9}{3} = 3 \]
4. Multiply the results:
\[ V = 36 \cdot 3 = 108 \]
Final Volume
The volume of the pyramid with a square base where each side is 6 units long and the height is 9 units is 108 cubic units.
