Calculating the volume of a pyramid with a rectangular base is a straightforward process when you use the appropriate formula. This guide will walk you through the steps, explain the necessary formulas, and provide a practical example to ensure you can find the volume of any rectangular-based pyramid.
Volume of a Pyramid with a Rectangular Base Formula
The volume (\( V \)) of a pyramid with a rectangular base can be determined using the formula:
\[ V = \dfrac{l \cdot w \cdot h}{3} \]
Where:
- \( l \) is the length of the rectangular base.
- \( w \) is the width of the rectangular base.
- \( h \) is the height of the pyramid.
Explanation of the Formula
- The term \(\dfrac{1}{3}\) represents the fact that the volume of a pyramid is one-third the volume of a prism with the same base area and height.
- \( l \cdot w \) is the area of the rectangular base.
- Multiplying the base area by the height (\( h \)) and then dividing by 3 gives the total volume of the pyramid.
Step-by-Step Calculation
Let’s go through an example to demonstrate how to use this formula to find the volume of a pyramid with a rectangular base.
Example: Calculating the Volume of a Pyramid with a Rectangular Base
1. Identify the given values:
- Length of the base (\( l \)) = 8 units
- Width of the base (\( w \)) = 5 units
- Height of the pyramid (\( h \)) = 10 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{8 \cdot 5 \cdot 10}{3} \]
3. Simplify the expression inside the formula:
\[ l \cdot w = 8 \cdot 5 = 40 \]
\[ 40 \cdot 10 = 400 \]
4. Substitute back into the formula:
\[ V = \dfrac{400}{3} \]
5. Complete the division:
\[ V \approx 133.33 \text{ cubic units} \]
Final Volume
The volume of the pyramid with a rectangular base of length 8 units, width 5 units, and height 10 units is approximately 133.33 cubic units.
