Finding the capacity of a hollow cylinder is essentially calculating its internal volume – the space inside the cylinder that can hold a substance. This is determined by the difference between the volume of the outer cylinder and the volume of the inner hollow space.
As discussed in educational resources like the video titled "Volume of a hollow cylinder" on YouTube, calculating the volume of a hollow cylinder involves applying the general principle of volume calculation, similar to finding the volume of a prism or a solid cylinder (Area of Base × Height).
Understanding the Components
A hollow cylinder has two main parts defining its capacity:
- Outer Cylinder: Defined by the outer radius (R) and the height (h).
- Inner Cylinder (The Hollow Space): Defined by the inner radius (r) and the same height (h).
The capacity is the volume enclosed by the inner surface of the cylinder, which is determined by the inner radius and height. However, the overall object's volume (if you were to melt it down, for example) would be the outer volume minus the inner volume. When discussing "capacity" in the sense of "how much can it hold," we are typically referring to the volume defined by the inner dimensions. Let's clarify this common ambiguity.
Interpretation 1: Capacity as the Volume the Hollow Space Can Hold (Internal Volume)
This is the most common interpretation of "capacity" for a container. It's the volume of the space inside.
Interpretation 2: Capacity as the Volume of the Material Making Up the Cylinder Wall (Solid Volume)
This is the outer volume minus the inner volume. This is what the linked video primarily calculates.
Since the video title is "Volume of a hollow cylinder" and discusses the calculation, it likely focuses on the volume of the material itself (Interpretation 2), as this is the unique calculation for a hollow shape versus a solid one. However, the term "capacity" usually implies Interpretation 1. Let's address both, starting with the volume calculation as shown in the video's topic.
Calculating the Volume of the Hollow Cylinder (Material Volume)
This method, which aligns with calculating the total volume of the hollow shape as discussed in the reference, involves subtracting the volume of the inner cylindrical space from the volume of the outer cylinder.
The formula for the volume of a solid cylinder is:
$$V = \pi \times radius^2 \times height$$
Where $\pi$ (Pi) is approximately 3.14159.
For a hollow cylinder:
- Volume of Outer Cylinder (V_outer): $V_{outer} = \pi \times R^2 \times h$ (where R is the outer radius)
- Volume of Inner Cylinder (V_inner): $V_{inner} = \pi \times r^2 \times h$ (where r is the inner radius)
The volume of the hollow cylinder material is the difference:
$$V{hollow} = V{outer} - V{inner}$$
$$V{hollow} = (\pi \times R^2 \times h) - (\pi \times r^2 \times h)$$
By factoring out $\pi$ and $h$, the formula simplifies to:
$$V_{hollow} = \pi \times (R^2 - r^2) \times h$$
This formula calculates the volume of the material that forms the walls of the hollow cylinder.
Calculating the Capacity (Internal Volume)
If "capacity" refers to how much the hollow cylinder can hold (like a pipe or a storage tank), you are interested in the volume of the inner cylindrical space.
This is simply the volume of the cylinder defined by the inner radius (r) and the height (h):
$$Capacity = V_{inner} = \pi \times r^2 \times h$$
This is the most practical definition of capacity for a container.
Step-by-Step Calculation
To find either the material volume or the internal capacity, you need specific measurements:
- Measure the Height (h): The vertical distance from the base to the top of the cylinder.
- Measure the Outer Radius (R) or Outer Diameter (D): If you measure the diameter (D), divide it by 2 to get the radius (R = D/2). This is the distance from the center to the outer edge.
- Measure the Inner Radius (r) or Inner Diameter (d): If you measure the diameter (d), divide it by 2 to get the radius (r = d/2). This is the distance from the center to the inner edge of the hollow space.
Note: The wall thickness (t) of the cylinder is the difference between the outer and inner radii: $t = R - r$. You can also find $R$ if you know $r$ and $t$ ($R = r + t$), or find $r$ if you know $R$ and $t$ ($r = R - t$).
Once you have these values, substitute them into the appropriate formula:
- For Material Volume: $V_{hollow} = \pi \times (R^2 - r^2) \times h$
- For Internal Capacity: $Capacity = \pi \times r^2 \times h$
Example Calculation
Let's find the capacity (internal volume) of a hollow cylinder with the following dimensions:
- Inner Radius (r) = 5 cm
- Outer Radius (R) = 6 cm
- Height (h) = 10 cm
Using the formula for Internal Capacity:
$$Capacity = \pi \times r^2 \times h$$
$$Capacity = \pi \times (5 \text{ cm})^2 \times 10 \text{ cm}$$
$$Capacity = \pi \times 25 \text{ cm}^2 \times 10 \text{ cm}$$
$$Capacity = 250\pi \text{ cm}^3$$
Using the approximation $\pi \approx 3.14159$:
$$Capacity \approx 250 \times 3.14159 \text{ cm}^3$$
$$Capacity \approx 785.3975 \text{ cm}^3$$
The internal capacity of this hollow cylinder is approximately 785.4 cubic centimeters.
If we were calculating the material volume (as discussed in the reference video's topic):
$$V{hollow} = \pi \times (R^2 - r^2) \times h$$
$$V{hollow} = \pi \times ((6 \text{ cm})^2 - (5 \text{ cm})^2) \times 10 \text{ cm}$$
$$V{hollow} = \pi \times (36 \text{ cm}^2 - 25 \text{ cm}^2) \times 10 \text{ cm}$$
$$V{hollow} = \pi \times (11 \text{ cm}^2) \times 10 \text{ cm}$$
$$V_{hollow} = 110\pi \text{ cm}^3$$
Using the approximation $\pi \approx 3.14159$:
$$V{hollow} \approx 110 \times 3.14159 \text{ cm}^3$$
$$V{hollow} \approx 345.5749 \text{ cm}^3$$
The volume of the material in the hollow cylinder is approximately 345.6 cubic centimeters.
In summary, while the reference discusses the volume of the hollow structure itself, the capacity (what it can hold) is specifically its internal volume calculated using the inner radius.
