Finding the volume of a pyramid with a triangular base involves using a specific formula that takes into account the area of the triangular base and the height of the pyramid. This article 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 Triangular Base Formula
To calculate the volume (\( V \)) of a pyramid with a triangular base, you can use the following formula:
\[ V = \dfrac{b \cdot h_b \cdot h}{2} \cdot \dfrac{1}{3} = \dfrac{b \cdot h_b \cdot h}{6} \]
Where:
- \( b \) is the base length of the triangular base.
- \( h_b \) is the height of the triangular base.
- \( h \) is the height of the pyramid (the perpendicular distance from the base to the apex).
Explanation of the Formula
- The term \(\dfrac{b \cdot h_b}{2}\) represents the area of the triangular base (since the area of a triangle is \(\dfrac{1}{2} \cdot \text{base} \cdot \text{height}\)).
- 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.
- Combining these, the formula becomes \(\dfrac{b \cdot h_b \cdot h}{6}\).
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 Triangular Base
1. Identify the given values:
- Base length of the triangular base (\( b \)) = 8 units
- Height of the triangular base (\( h_b \)) = 5 units
- Height of the pyramid (\( h \)) = 12 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{8 \cdot 5 \cdot 12}{6} \]
3. Simplify the expression inside the formula:
\[ 8 \cdot 5 = 40 \]
\[ 40 \cdot 12 = 480 \]
\[ \dfrac{480}{6} = 80 \]
Final Volume
The volume of the pyramid with a triangular base where the base length is 8 units, the height of the triangular base is 5 units, and the height of the pyramid is 12 units is 80 cubic units.
