Calculating the volume of a right truncated cylinder can be achieved using a specific formula that considers the radii of the two circular bases and the height of the cylinder. This article will guide you through the steps to find the volume using the appropriate formula. We'll explain the formula, show an example, and provide the final value.
Understanding the Volume Formula
The volume (\( V \)) of a right truncated cylinder can be calculated using the following formula:
\[ V = \dfrac{1}{2} \cdot \pi \cdot r^2 \cdot (h_1 + h_2) \]
Where:
- \( r \) is the radius of the cylinder.
- \( h_1 \) is the smaller height of the truncated part.
- \( h_2 \) is the larger height of the truncated part.
- \( \pi \) (Pi) is a constant approximately equal to 3.14159.
Explanation of the Formula
- The term \( r^2 \) represents the area of the circular base.
- Multiplying by \( \pi \) gives the exact area of the circular base.
- The term \( \dfrac{1}{2} \cdot (h_1 + h_2) \) calculates the average height of the truncated cylinder, effectively treating it as an average height times the base area.
Step-by-Step Calculation
Let's calculate the volume of a right truncated cylinder with given dimensions.
Example: Calculating the Volume of a Right Truncated Cylinder
1. Identify the given values:
- Radius of the base (\( r \)) = 5 units
- Smaller height (\( h_1 \)) = 8 units
- Larger height (\( h_2 \)) = 12 units
2. Substitute the values into the volume formula:
\[ V = \dfrac{1}{2} \cdot \pi \cdot r^2 \cdot (h_1 + h_2) \]
\[ V = \dfrac{1}{2} \cdot \pi \cdot 5^2 \cdot (8 + 12) \]
3. Calculate the area of the base:
\[ 5^2 = 25 \]
4. Sum the heights:
\[ 8 + 12 = 20 \]
5. Calculate the volume:
\[ V = \dfrac{1}{2} \cdot \pi \cdot 25 \cdot 20 \]
6. Simplify the expression:
\[ V = \dfrac{1}{2} \cdot 500 \cdot \pi \]
7. Use the value of \( \pi \approx 3.14159 \) to get the final volume:
\[ 250 \cdot \pi \approx 250 \cdot 3.14159 = 785.398 \]
Final Value
The volume of a right truncated cylinder with a radius of 5 units, a smaller height of 8 units, and a larger height of 12 units is approximately 785.40 cubic units.
