Calculating the volume of a hollow right cylinder is a straightforward process if you understand the relationship between its outer radius, inner radius, and height. 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 hollow right cylinder can be calculated using the following algebraic formula:
\[ V = \pi \cdot h \cdot (R^2 - r^2) \]
Where:
- \( R \) is the outer radius of the cylinder.
- \( r \) is the inner radius of the cylinder.
- \( h \) is the height of the cylinder.
- \( \pi \) (Pi) is a constant approximately equal to 3.14159.
Explanation of the Formula
- The term \( R^2 \) represents the area of the outer circular base.
- The term \( r^2 \) represents the area of the inner circular base.
- Subtracting \( r^2 \) from \( R^2 \) gives the area of the annular (ring-shaped) cross-section.
- Multiplying by \( \pi \) gives the exact area of the annular cross-section.
- Multiplying by \( h \) extends this annular area through the height of the cylinder, resulting in the total volume.
Step-by-Step Calculation
Let's calculate the volume of a hollow right cylinder with given dimensions.
Example: Calculating the Volume of a Hollow Right Cylinder
1. Identify the given values:
- Outer radius of the cylinder (\( R \)) = 5 units
- Inner radius of the cylinder (\( r \)) = 3 units
- Height of the cylinder (\( h \)) = 7 units
2. Substitute the values into the volume formula:
\[ V = \pi \cdot h \cdot (R^2 - r^2) \]
\[ V = \pi \cdot 7 \cdot (5^2 - 3^2) \]
3. Calculate the areas of the outer and inner bases:
\[ 5^2 = 25 \]
\[ 3^2 = 9 \]
4. Subtract the inner area from the outer area:
\[ 25 - 9 = 16 \]
5. Calculate the volume:
\[ V = \pi \cdot 7 \cdot 16 \]
\[ V = 112\pi \]
6. Use the value of \( \pi \approx 3.14159 \) to get the final volume:
\[ 112\pi \approx 112 \cdot 3.14159 = 351.85888 \]
Final Value
The volume of a hollow right cylinder with an outer radius of 5 units, an inner radius of 3 units, and a height of 7 units is approximately 351.86 cubic units.
Summary
By using the formula \( V = \pi \cdot h \cdot (R^2 - r^2) \), you can easily calculate the volume of a hollow right cylinder. This fundamental geometric concept is essential for various academic and practical applications.
