To find the volume of a hollow rectangular prism, you calculate the volume of its outer dimensions and subtract the volume of its inner, empty space. This method determines the volume of the material that makes up the prism.
Understanding a Hollow Rectangular Prism
A hollow rectangular prism is essentially a box-like structure with a void inside, often with walls of uniform thickness. Imagine a rectangular container or a pipe with a square or rectangular cross-section. It has both external (outer) and internal (inner) dimensions.
- Outer Dimensions: These define the total space the object occupies.
- Inner Dimensions: These define the empty space or cavity within the object.
- Thickness: The distance between the outer and inner surfaces.
The Core Formula: Volume of Material
The most common interpretation for the "volume of a hollow rectangular prism" refers to the volume of the solid material itself.
To calculate the volume of any rectangular prism, whether it's a solid block or an empty cavity, we use the fundamental formula:
$$ \text{Volume (V)} = \text{Length (L)} \times \text{Width (W)} \times \text{Height (H)} $$
This principle is applied twice: once for the outer dimensions and once for the inner dimensions.
Step-by-Step Calculation
Here’s how to determine the volume of the material:
- Calculate the Outer Volume ($V_{outer}$):
- Use the prism's external length ($L_o$), external width ($W_o$), and external height ($H_o$).
- $$ V_{outer} = L_o \times W_o \times H_o $$
- Calculate the Inner Volume ($V_{inner}$):
- Use the prism's internal length ($L_i$), internal width ($W_i$), and internal height ($H_i$). This represents the volume of the hollow space.
- $$ V_{inner} = L_i \times W_i \times H_i $$
- Calculate the Volume of the Material ($V_{material}$):
- Subtract the inner volume from the outer volume.
- $$ V{material} = V{outer} - V_{inner} $$
For a deeper understanding of basic volume calculations, you can refer to resources like Khan Academy's explanation of volume.
Determining Inner Dimensions from Outer Dimensions and Thickness
Often, you'll be given the outer dimensions and the uniform thickness of the material. To find the inner dimensions, you must account for the thickness on both sides of each dimension.
Let:
- $L_o$, $W_o$, $H_o$ be the outer length, width, and height.
- $t$ be the uniform thickness of the material.
Then, the inner dimensions are:
- Inner Length ($L_i$): $L_i = L_o - 2t$
- Inner Width ($W_i$): $W_i = W_o - 2t$
- Inner Height ($H_i$): $H_i = H_o - 2t$
Important Note: If the thickness is not uniform, or if the hollow space doesn't have the same rectangular shape as the outer prism (e.g., a complex internal structure), more advanced methods or detailed internal measurements would be required. However, for a typical "hollow rectangular prism," uniform thickness is assumed.
Practical Example
Let's find the volume of the material for a hollow rectangular prism.
Given:
- Outer Length ($L_o$) = 10 cm
- Outer Width ($W_o$) = 6 cm
- Outer Height ($H_o$) = 5 cm
- Material Thickness ($t$) = 1 cm
Step 1: Determine Inner Dimensions
Using the thickness:
- $L_i = 10 \text{ cm} - (2 \times 1 \text{ cm}) = 10 \text{ cm} - 2 \text{ cm} = 8 \text{ cm}$
- $W_i = 6 \text{ cm} - (2 \times 1 \text{ cm}) = 6 \text{ cm} - 2 \text{ cm} = 4 \text{ cm}$
- $H_i = 5 \text{ cm} - (2 \times 1 \text{ cm}) = 5 \text{ cm} - 2 \text{ cm} = 3 \text{ cm}$
Here’s a summary of the dimensions:
| Dimension | Outer Value (cm) | Inner Value (cm) |
|---|---|---|
| Length | 10 | 8 |
| Width | 6 | 4 |
| Height | 5 | 3 |
Step 2: Calculate Outer Volume
Using the outer dimensions:
- $V_{outer} = L_o \times W_o \times H_o$
- $V_{outer} = 10 \text{ cm} \times 6 \text{ cm} \times 5 \text{ cm}$
- $V_{outer} = 300 \text{ cm}^3$
Step 3: Calculate Inner Volume
Using the inner dimensions:
- $V_{inner} = L_i \times W_i \times H_i$
- $V_{inner} = 8 \text{ cm} \times 4 \text{ cm} \times 3 \text{ cm}$
- $V_{inner} = 96 \text{ cm}^3$
Step 4: Calculate Volume of Material
Subtract the inner volume from the outer volume:
- $V{material} = V{outer} - V_{inner}$
- $V_{material} = 300 \text{ cm}^3 - 96 \text{ cm}^3$
- $V_{material} = 204 \text{ cm}^3$
Therefore, the volume of the material in this hollow rectangular prism is 204 cubic centimeters.
Other Interpretations of "Volume"
While the volume of the material is the most common meaning, sometimes "volume of a hollow rectangular prism" might implicitly refer to:
- Total Enclosed Volume: This would simply be the outer volume ($V_{outer}$), representing the entire space the object occupies, including the hollow part. In our example, this would be 300 cm³.
- Internal Capacity/Volume: This refers to the volume of the empty space inside the prism ($V_{inner}$), which is the maximum amount of substance (liquid, gas, etc.) it can hold. In our example, this would be 96 cm³.
It is crucial to clarify which "volume" is being sought based on the context of the problem.
Why Is This Important?
Calculating the volume of the material of a hollow prism has practical applications in many fields:
- Engineering and Manufacturing: Determining the amount of raw material needed (e.g., plastic for a container, metal for a frame), calculating weight, or understanding structural properties.
- Architecture and Construction: Estimating the volume of concrete for hollow blocks or pipes.
- Packaging: Designing boxes with specific internal capacities while considering the material volume for shipping weight.
Understanding how to accurately calculate these volumes is fundamental for efficient design, costing, and resource management.
