To find the capacity of a hollow cylinder, you calculate its volume, which represents the amount of substance it can hold or displace. This is achieved by determining the volume of the material that makes up the cylinder, or the space enclosed within its outer boundaries, minus any inner void.
Understanding Hollow Cylinder Capacity
The capacity of a hollow cylinder is synonymous with its volume. It measures the three-dimensional space occupied by the cylinder's material. Unlike a solid cylinder, a hollow cylinder has an inner void, meaning its capacity depends on the difference between its outer and inner dimensions.
The Formula for Hollow Cylinder Volume (Capacity)
Based on the provided reference, the formula for the volume (capacity) of a hollow cylinder is:
$$ \mathbf{V = \pi (R^2 - r^2)h \text{ cubic units}} $$
Where:
- V represents the volume or capacity of the hollow cylinder.
- π (Pi) is a mathematical constant approximately equal to 3.14159.
- R is the outer radius of the cylinder.
- r is the inner radius of the cylinder.
- h is the height of the cylinder.
This formula essentially calculates the volume of the larger (outer) cylinder and subtracts the volume of the smaller (inner) cylinder, leaving only the volume of the material or the space between the inner and outer surfaces.
Components of the Formula
Let's break down the key variables and their significance:
| Variable | Description | Unit of Measurement (Example) |
|---|---|---|
| V | Volume / Capacity | cubic meters (m³), liters (L) |
| π | Mathematical constant (approx. 3.14159) | Unitless |
| R | Outer Radius (distance from center to outer edge) | meters (m), centimeters (cm) |
| r | Inner Radius (distance from center to inner edge) | meters (m), centimeters (cm) |
| h | Height (length of the cylinder) | meters (m), centimeters (cm) |
Practical Applications and Examples
Understanding the capacity of a hollow cylinder is crucial in various fields, including engineering, construction, and manufacturing.
- Piping Systems: Calculating the volume of water a pipe can hold (its internal capacity) or the material needed to manufacture the pipe itself (its material volume).
- Storage Tanks: Determining the storage capacity of cylindrical tanks with internal structures.
- Structural Elements: Assessing the strength and material requirements for hollow columns or beams.
Step-by-Step Calculation Example
Let's find the capacity of a hollow cylinder with the following dimensions:
- Outer Radius (R) = 10 cm
- Inner Radius (r) = 8 cm
- Height (h) = 30 cm
-
Identify the given values:
- R = 10 cm
- r = 8 cm
- h = 30 cm
-
Apply the formula:
V = π (R² - r²)h -
Substitute the values:
V = π ((10 cm)² - (8 cm)²) 30 cm
V = π (100 cm² - 64 cm²) 30 cm
V = π (36 cm²) * 30 cm
V = 1080π cm³ -
Calculate the numerical value (using π ≈ 3.14159):
V ≈ 1080 * 3.14159 cm³
V ≈ 3392.92 cm³
Therefore, the capacity of this hollow cylinder is approximately 3392.92 cubic centimeters.
Key Takeaways
- The capacity of a hollow cylinder is its volume.
- The formula accounts for the inner void by subtracting the inner cylinder's volume from the outer cylinder's volume.
- Ensure all measurements (R, r, h) are in the same units to obtain the volume in consistent cubic units.
For more information on geometric calculations, you might explore resources on Cylinder Volume Calculators.
