Volume of a Rectangular Prism
Introduction
A rectangular prism is a polyhedron with two congruent and parallel bases. A rectangular prism is also termed a cuboid. A rectangular prism has six faces and twelve edges. All the six faces are rectangular in shape. Volume is the total amount of space a `3D` object occupies. The total space inside a rectangular prism is called its volume. Picture a rectangular tank that is full of water. In this instance, the capacity of the tank equals the entire amount of water it can hold. Since the rectangular prism is a three-dimensional `(3D)` shape, its volume is expressed in cubic units like `\text{cm}^3`, `\text{m}^3` etc. In this article, we will study the formula for calculating a rectangular prism's volume.
What Do You Mean by Volume of a Rectangular Prism?
The volume of a rectangular prism, also visualized as a rectangular solid or box, refers to the amount of space enclosed by the prism. It represents the total capacity or the total amount of substance that the prism can hold.
A rectangular prism is a three-dimensional geometric shape characterized by six faces, each of which is a rectangle. These rectangles form three pairs of congruent opposite faces. The length, width, and height of the rectangular prism determine its dimensions. The volume of a rectangular prism is expressed in cubic units, such as cubic centimeters `(\text{cm}^3)` or cubic meters `(\text{m}^3)`, depending on the units used for the dimensions.
It is a fundamental concept in geometry and is widely used in various real-world applications, such as calculating the capacity of containers, determining the amount of material needed for construction or manufacturing, and solving problems in engineering and physics.
Formula for Finding Volume of a Rectangular Prism
The rectangular prism volume formula for finding the volume (\(V\)) of a rectangular prism is:
\( V = l \times w \times h \)
where
- \( l \) is the length of the rectangular prism.
- \( w \) is the width of the rectangular prism.
- \( h \) is the height of the rectangular prism.
We multiply the length (\(l\)) by the width (\(w\)) and then by the height (\(h\)) to find the total amount of space enclosed by the prism. This formula represents the product of the three dimensions of the rectangular prism, resulting in a rectangular prism volume measured in cubic units.
We can find numerous things around us that are rectangular prism like a cereal box, a show box, a chest of drawers, a laptop, books, aquarium, etc.
Applications of Volume of a Rectangular Prism
`1`. Packaging and Shipping: Companies use the volume of rectangular prisms to determine the amount of space needed to package and ship products efficiently. Understanding volume helps optimize packaging design and transportation logistics, leading to cost savings and reduced environmental impact.
`2`. Storage Capacity: The volume of a rectangular prism is crucial for determining the storage capacity of containers, warehouses, and storage units. It helps businesses and individuals plan and organize storage space effectively, maximizing utilization while minimizing waste.
`3`. Construction and Architecture: Architects and engineers use volume calculations to design buildings, rooms, and structures. Understanding the volume of materials required, such as concrete, steel, or lumber, helps in estimating costs, planning construction projects, and ensuring structural integrity.
`4`. Manufacturing: Volume calculations are essential in manufacturing processes for determining the quantity of raw materials needed to produce goods. Manufacturers use volume to optimize production efficiency, minimize waste, and maintain consistent quality standards.
`5`. Fluid Mechanics: In fluid dynamics and hydraulic engineering, volume calculations play a vital role in designing pipes, tanks, and reservoirs for storing and transporting liquids. Understanding volume helps ensure proper flow rates, pressure levels, and fluid distribution in hydraulic systems.
`6`. Education and Learning: Volume concepts are taught in mathematics and science education at various levels. Students learn how to calculate volume to solve real-world problems, develop spatial reasoning skills, and understand geometric principles.
`7`. Medical Imaging: In medical imaging technologies such as CT scans and MRI scans, volume measurements are used to quantify the size and shape of organs, tumors, and other anatomical structures. Accurate volume calculations aid in diagnosis, treatment planning, and monitoring of medical conditions.
`8`. Environmental Studies: Volume calculations are used in environmental science and ecology to measure the volume of natural features such as lakes, rivers, and forests. Understanding volume helps researchers assess ecosystem health, monitor changes over time, and develop conservation strategies.
Solved Examples
Example `1`: What is the volume of rectangular prism with the following dimensions:
Length (\( l \)) `= 5` cm,
Width (\( w \)) `= 3` cm,
Height (\( h \)) `= 4` cm.
Solution:
\( V = l \times w \times h \)
\( V = 5 \times 3 \times 4 \)
\( V = 60 \text{ cubic centimeters} \)
So, the volume of the rectangular prism is \( 60 \) cubic centimeters.
Example `2`: A rectangular prism has a volume of \( 240 \) cubic meters. If it is `8` meters long and `5` meters wide, what is the height of the prism?
Solution:
Given: \( V = 240 \) cubic meters, \( l = 8 \) meters, \( w = 5 \) meters.
\( V = l \times w \times h \)
\( 240 = 8 \times 5 \times h \)
\( 240 = 40h \)
`h = \frac{240}{40}`
\( h = 6 \text{ meters} \)
So, the height of the rectangular prism is \( 6 \) meters.
Example `3`: The area of the base of a rectangular prism is `45` square inches. If the volume of the prism is `225` cubic meters, what is its height?
Solution:
Given: \( V = 225 \) cubic inches, \( a = 45 \) square inches, \( w = 5 \) inches.
Here `a` is the base area which is \( l \times w \).
\( V = l \times w \times h \)
\( 225 = a \times h \)
\( 225 = 45h \)
`h = \frac{225}{45}`
\(h = 5 \text{ inches} \)
So, the height of the rectangular prism is \( 5 \) inches.
Example `4`: Find the combined volume of the rectangular prism shown in the composite figure below:
Solution:
In the composite figure, we have a tall rectangular prism attached to a short rectangular prism.
Let’s first deal with the long rectangular prism.
\( l = 6 \) in, \( w = 5 \) in and \( h = 12 \) in.
\( V_1 = l \times w \times h \)
\( V_1 = 6 \times 5 \times 12 \)
\( V_1 = 360 \text{ cubic inches} \)
Now let’s look into the short rectangular prism.
\( l = 14 - 6 = 8 \) in, \( w = 5 \) in and \( h = 12 - 8 = 4 \) in.
\( V_2 = l \times w \times h \)
\( V_2 = 8 \times 5 \times 4 \)
\( V_2 = 160 \text{ cubic inches} \)
So the combined volume of the composite figure is \(V_1 + V_2\) which is \(360 + 160 = 520 \text{ cubic inches}\).
Example `5`: The volume of a rectangular tank is \( 800 \) cubic centimeters. If the length of the tank is reduced to `frac{2}{3}` rd of its original length and the height is tripled, what would be the new volume?
Solution:
Let the original length, width and height of the tank be `l`, `w`, and `h` respectively.
Hence `lwh = 800`
New length of the tank `=\frac{2}{3}l`.
New height of the tank `= 3h`.
New volume of the tank `=\frac{2}{3}l \times w \times 3h`
`=\frac{2}{3} \times 3 \times lwh`
\( =2 \times lwh \)
Substituting `lwh = 800`, we get
New volume `=2(800) = 1600` cubic centimeters.
Practice Problems
Q`1`. What is the formula for finding the volume (\(V\)) of a rectangular prism?
- \(V = l + w + h\)
- \(V = l \times w \times h\)
- `V = \frac{l}{w} \times h`
- `V = \frac{l + w}{h}`
Answer: b
Q`2`. A rectangular prism has a length (\(l\)) of `6` meters, a width (\(w\)) of `4` meters, and a height (\(h\)) of `3` meters. What is its volume?
- `36` cubic meters
- `48` cubic meters
- `72` cubic meters
- `96` cubic meters
Answer: c
Q`3`. If the volume of a rectangular prism is `400` cubic inches and its length is `10` inches, what is its width if the height is `5` inches?
- `4` inches
- `8` inches
- `16` inches
- `20` inches
Answer: b
Q`4`. Find the base area of a rectangular prism whose volume is `345` cubic inches and height is `15` inches.
- `23` centimeters
- `81` centimeters
- `32` centimeters
- `67` centimeters
Answer: a
Q`5`. The volume of a rectangular tank is \( 550 \) cubic centimeters. If the length of the tank is doubled and the width is tripled, what would be the new volume?
- `3000` cubic feet
- `3300` cubic feet
- `2800` cubic feet
- `4000` cubic feet
Answer: b
Frequently Asked Questions
Q`1`. What is the volume of a rectangular prism?
Answer: The volume of a rectangular prism refers to the amount of space enclosed by the prism. It is calculated by multiplying the length, width, and height of the prism.
Q`2`. How to find the volume of a rectangular prism?
Answer:To find the volume of a rectangular prism, you multiply its length, width, and height together. The formula for volume of a rectangular prism (\(V\)) is: \(V = l \times w \times h\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height.
Q`3`. What are the units of volume used for rectangular prisms?
Answer: The units of volume depend on the units used for the dimensions of the rectangular prism. Common units of volume include cubic centimeters `(\text{cm}^3)`, cubic meters `(\text{m}^3)`, cubic inches `(\text{in}^3)`, and cubic feet `(\text{ft}^3)`.
Q`4`. Why is volume important in real-life applications?
Answer: Volume calculations are essential in various fields and everyday tasks. They are used in construction, manufacturing, packaging, transportation, and many other applications to determine capacity, quantify amounts of materials, optimize space, and solve practical problems.
Q`5`. Can the volume of a rectangular prism be negative?
Answer: No, the volume of a rectangular prism cannot be negative. Volume represents a measure of space, and it is always a positive quantity or zero. If the length, width, or height of a rectangular prism is negative, it would indicate an error in measurement or calculation rather than a negative volume.