Volume is easily defined as the amount of space occupied by any three-dimensional solid. These solids can be cubes, cuboids, cones, cylinders, or globes.
Different shapes have different volumes. In 3-D geometry, we have studied various shapes and solid objects such as cubes, cylinders, cones, cuboids and so on, described in three dimensions.
What is a Rectangular Prism?
A rectangular prism is a three dimensional shape of a hexagon (2 tops, 2 bottoms, 4 sides). All of the prism’s faces are rectangular. As a result, there are three pairs of identical faces in this picture. A rectangular prism is also known as a cuboid because of its shape. A rectangular prism can be found in a geometry box, notebooks, diaries, rooms, and other places. The shape of a rectangular prism can be seen in the diagram below.
Examples of Rectangular Prisms in Real Life
A rectangular prism can be found in a truck, a chest of drawers, and an aquarium, among other place
Formula for the Volume of a Rectangular Prism
The occupied units of a rectangular prism are measured by the volume of the prism. Cubic units are used to represent the volume of a rectangular prism. The number of units used to fill a rectangular prism is also defined.
The area of the base multiplied by the height equals the volume of the rectangular prism.
A rectangular prism’s volume is equal to the multiplication of its length, width, and height in cubic units.
=l× w× h
cubic units of volume
Where l is the length, w is the width, and h is the height of the rectangular prism.
How do You Calculate the Volume of a Rectangular Prism
Before using the formula to calculate the volume of a rectangular prism, make sure that all of the dimensions are in the same units. The volume of a rectangular prism is calculated using the steps below.
Step 1: Determine the type of base and its area using an appropriate formula (as explained in the previous section).
Step 2: Determine the prism’s height, which is perpendicular to the prism’s top vertex and base.
Step 3: To calculate the volume of the rectangular prism in cubic units, multiply the base area by the height of the prism. The volume of a prism is equal to the base area multiplied by the prism’s height.
Volume of Rectangular Prism Examples
Example 1:
Calculate the volume of a rectangular prism with dimensions of 8 ft, 5 ft, and 4 ft, respectively.
Solution:
Given that:
Length, l=8ft
Width, w=5ft
Height, h=4ft
The formula for calculating the volume of a rectangular prism is as follows:
Volume = Length x Width x Height cubic units
Volume = 8× 5×4
cubic feet
Volume = 160 cubic feet
As a result, a rectangular prism’s volume is 160 cubic feet.
Example 2:
The bed of a dump truck is shaped like a rectangular prism. The base is 8 square yards in size. It stands at a height of 2 yards. How much gravel can it transport at once?
Solution:
Use a formula to calculate the volume of the dump truck bed.
Volume = Base area × height [Formula for volume of a rectangular prism]
= 8 × 2
= 16 cubic yards
The volume of the dump truck bed is 16 cubic yards.
So, the driver can transport 16 cubic yards of gravel at a time.
Example 3:
The volume of the rectangular prism is 72 cubic feet. Find the height.
Example 4:
The candle has a volume of 640 cubic centimetres. Will the candle fit inside the glass jar completely? Explain.
The height of the candle is 16 feet.
Compare the base area and height of the candle and the glass jar:
Base area of the glass jar is greater than the candle.
And the height of the glass jar is less than the candle’s height.
So, the candle cannot fit inside the glass jar completely.
The volume of a rectangular prism refers to the amount of material it can hold or the amount of space it takes up. A rectangular prism’s volume can thus be calculated by multiplying its base area by its height.
Volume (V) = height of the prism × base area is the formula for calculating the volume of a rectangular prism.
What is the formula for finding the volume of a rectangular prism?
Volume (V) = Area of base × height of the prism is the formula for calculating the volume of a rectangular prism.
Volume = l × w × h is another way to express this formula, where ‘l’ is the length, ‘w’ is the width, and ‘h’ is the prism’s height.
What is the unit of volume?
Volume is measured in cubic units such as cubic feet, cubic inches, cubic yards, and so on.
Volume of Rectangular Prism
The volume of a rectangular prism is the measurement of the total space inside it. Imagine a rectangular container filled with water. In this case, the total quantity of water that the container can hold is its volume. A prism is a polyhedron having identical bases, flat rectangular side faces, and the same cross-section all along its length. Prisms are classified as per the shape of their base. A rectangular prism is categorized as a three-dimensional shape. It has six faces and all the faces of the prism are rectangles. Let us learn the formula to find the volume of a rectangular prism in this article.
What is the Volume of Rectangular Prism?
The volume of a rectangular prism is defined as the space occupied within a rectangular prism. A rectangular prism is a polyhedron that has two pairs of congruent and parallel bases. It has 6 faces (all are rectangular),12 sides, and 8 vertices. As the rectangular prism is a three-dimensional shape (3D shape), the unit that is used to express the volume of the rectangular prism is cm3, m3 and so on. In mathematics, any polyhedron, having all such characteristics can be referred to as a Cuboid.
Volume of Rectangular Prism Formula
The formula for the volume of a rectangular prism = base area × height of the prism. Since the base of a rectangular prism is a rectangle, its area will be l × w. This area is then multiplied by the height of the prism to get the volume of the prism. Therefore, another way to express this formula is by multiplying the length, width, and height of the prism and write the value in cubic units (cm3, m3, in3, etc).
Therefore, the formula for the volume of a rectangular prism is, volume of a rectangular prism (V) = l × w × h, where
- “l” is the base length
- “w” is the base width
- “h” is the height of the prism
To find the volume of the rectangular prism, we multiply the length, width, and height, that is the area of the base is multiplied by the height. It should be noted that the volume is measured in cubic units.
There are two types of rectangular prisms – right rectangular prisms and oblique prisms.
- In the case of a right rectangular prism, the bases are perpendicular to the other faces.
- In the case of an oblique rectangular prism, the bases are not perpendicular to the other faces. Thus, the perpendicular drawn from the vertex of one base to the other base of the prism will be taken as its height.
It should be noted that we can apply the same formula to calculate the volume of the prism, that is the rectangular prism volume formula, v = lwh, irrespective of the type of rectangular prism.
How to Find the Volume of a Rectangular Prism?
Before calculating the volume of a rectangular prism using the formula, we need to make sure that all the dimensions are of the same units. The following steps are used to calculate the volume of a rectangular prism.
- Step 1: Identify the type of the base and find its area using a suitable formula (as explained in the previous section).
- Step 2: Identify the height of the prism, which is perpendicular from the top vertex to the base of the prism.
- Step 3: Multiply the base area and the height of the prism to get the volume of the rectangular prism in cubic units. Volume = base area × height of the prism
Example: Calculate the volume of a rectangular prism whose height is 8 in and whose base area is 90 square inches.
Solution: We can calculate the volume of the rectangular prism using the following steps:
- Step 1: The base area is already given as 90 square inches.
- Step 2: The height of the prism is 8 in.
- Step 3: The volume of the given rectangular prism = base area × height of the prism = 90 × 8 = 720 cubic inches.
Volume of Rectangular Prism Examples
Example 1: If the volume of a rectangular prism is 40 cubic units and its base area is 10 square units, what is its height?
Solution: Given, the volume of the rectangular prism = 40 cubic units; and the base area of the rectangular prism= l × b = 10 square units.
Using the volume of rectangular prism formula, base area × height of the prism = 40
The height of the given rectangular prism = 10 × height = 40.
Thus, the height of the rectangular prism = 40/10 = 4 units
Example 2: If the base length of a rectangular prism is 8 inches, the base width is 5 inches and the height of the prism is 16 inches, find the volume of the rectangular prism.
Solution: Given: The base length of the rectangular prism (l) = 8 in, base width (w) = 5 in.
So the base area = l × w = 8 × 5 = 40 sq inches.
The height of the prism is h = 16 in.
Using the volume of the rectangular prism formula, the volume of rectangular prism = base area × height of the prism = (40 × 16) = 640 cubic inches.
FAQs on Volume of Rectangular Prism
What is the Volume of a Rectangular Prism?
The volume of a rectangular prism is the capacity that it can hold or the space occupied by it. Thus, the volume of a rectangular prism can be calculated by multiplying its base area by its height. The formula that is used to find the volume of a rectangular prism is, Volume (V) = height of the prism × base area. It is expressed in cubic units such as cm3, m3, in3, etc.
What is the Formula for the Volume of a Rectangular Prism?
The formula for the volume of a rectangular prism is, Volume (V) = base area × height of the prism. Another way to express this formula is, Volume = l × w × h; where ‘l’ is the length, ‘w’ is the width, and ‘h’ is the height of the prism.
How to Find the Volume of a Rectangular Prism?
In order to find the volume of a rectangular prism, follow the steps given below:
- Step 1: Identify the length (l) and width (w) of the base of the prism.
- Step 2: Find the height (h) of the prism.
- Step 3: Substitute the respective values in the formula, V = lwh
- Step 4: Find the product of these values to get the volume of the prism in cubic units.
How to Find the Height of a Rectangular Prism when the Volume of a Rectangular Prism is given?
The height of a rectangular prism can be calculated if the volume of the prism is given using the same formula, Volume (V) = base area × height of the prism. For example, the volume of a prism is 600 cubic units and its base area is 60 square units. Then, after substituting the values in the equation, we get 600 = 60 × height of the prism. This means, height = 600/60 = 10. Therefore, the height is 10 units.
What Happens to the Volume of Rectangular Prism if its Height is Doubled?
We know that the volume of a rectangular prism is the product of its three dimensions, that is, volume = length × width × height. If its height is doubled, its volume will be l × w × (2h) = 2lwh = 2 × v. Thus, we can say that the volume of the rectangular prism also gets doubled when its height is doubled.
What Happens to the Volume of Rectangular Prism if the Length is Doubled and the Height is Reduced to One Half?
The formula for calculating the volume of a rectangular prism is volume = length × width × height. If its length is doubled, and the height is reduced to half, then its volume can be written as, V = (2l) × (w) × (1/2h). After simplifying this, we get l × w × h which is the regular formula for volume. Thus, we can say that the volume of the rectangular prism will remain the same if its length gets doubled and the height is reduced to half.
What Happens to the Volume of Rectangular Prism if the Length, Width, and Height of Prism are Doubled?
The volume of a rectangular prism is the product of its three dimensions, that is volume = length × width × height. If its length, width, and height is doubled, then its volume will be (2l) × (2w) × (2h) = 8lwh = 8 × v. Thus, we can conclude that if its length, width, and height is doubled, the volume of the rectangular prism will be 8 times the original value.
How to Find the Volume of a Rectangular Prism with Fractions?
The volume of a rectangular prism can be calculated even if the values are given in fractions. For example, if the length of a
How is a rectangular prism different from a rectangle?
To begin with, why is it important to know the difference between different types of shapes?
Every shape has distinct properties and these properties help to know quantities such as volume, surface area, etc. Remember, you would not know to not put a rectangle on top of a triangle if you don’t know how they are different.
Difference Between Rectangle & Rectangular Prism
| Rectangle | Rectangular Prism |
| It is a 2D shape. | It is a 3D shape. |
| It has four sides. | It has six faces, eight verticals & twelve edges. |
| It is made of four sides with the opposite sides having equal lengths. | It is made up of six rectangles put together. |
| It has width and length. | It has width, height and length. |
| It is made of two pairs of lines. | It is made of three pairs of rectangles. |
Now that we know what rectangular prisms are, let’s look at how we can calculate its volume.
The Formula for Volume of a Rectangular Prism
By multiplying the base area of a prism by its height, you will get the volume of a prism. That is to say, the volume of a prism = base area × height.
Since a rectangular prism’s base is a rectangle itself, the volume of a rectangular prism, by applying the formula given above, will be:
Volume of a rectangular prism (V) = l × w × h
Where,
“l” means the base length, and
“w” means the base width,
“h” means the height
Solved Examples
Example 1: Find out the volume of a rectangular prism with base length 9 inches, base width 6 inches, and height 18 inches, respectively.
Solution:
Here,
Length (l) = 9 inches
Width (b) = 6 inches
Height (h) = 18 inches
So, the volume of the given rectangular prism = l × w × h = 9 × 6 × 18 = 972 cubic inches.
Example 2: Find out the height of a rectangular prism whose base area is 20 sq. units and a volume is 60 cubic units.
Solution:
Given is the base area of the rectangular prism = 20 sq. units
And the volume of the rectangular prism = 60 cubic units
Now, applying the volume of the rectangular prism formula,
base area × height = 60 cubic units.
⇒ 20 × height = 60
⇒ height = 60 ÷ 20 units
⇒ height = 3 units
Example 3: Find out the base area of a rectangular prism with the help of the given measurements: length = 12 inches, height = 20 inches, and volume = 2,160 cubic inches.
Solution:
Here,
Length (l) = 12 inches
Height (h) = 20 inches
Volume (V) = 2,160 cubic inches
The volume of the rectangular prism = l × w × h
⇒ 2,160 = 12 × w × 20
⇒ 2,160 ÷ (12 × 20) = w
⇒ 9 = w
Therefore, width (w) = 9 inches.
Area = l × w = 12 × 9 = 108 sq. inches.
Frequently Asked Questions
What is another name for a rectangular prism?
A rectangular prism is also known as a cuboid.
Is the volume of a rectangular prism the same as a cuboid?
Yes, the volume of a rectangular prism is the same as the volume of a cuboid.
What is an oblique rectangular prism?
An oblique rectangular prism is a prism with six rectangular faces, but the lateral faces are not perpendicular to the bases.
What is the surface area of a rectangular prism?
The surface area of a rectangular prism is the sum of the area of each of its faces. It is given by the formula, 2(lw + wh + hl), where l is length, w is width, and h is the height of the rectangular prism.
How do I calculate the volume of a rectangular prism?
To find the volume of a rectangular prism, you need to:
- Determine the lengths of the sides: width, length, and height.
- Multiply together the three values from Step 1.
- The result you’ve got is the volume of your solid.
Don’t forget to include the units if it is given! Since it’s volume, you need cubic units.
What is the volume of a rectangular prism with sides 2, 5, and 7?
The answer is 70. To see how to get this result, recall the formula for the volume of a rectangular prism:
volume = length × height × width
Hence, we compute the volume as 2 × 5 × 7 = 70.
Remember to include the units: for instance, if your measurements are in inches (in), the volume will be in cubic inches (in³).
How do I calculate the volume of a rectangular prism given diagonals?
To determine the volume of a rectangular prism when you know the diagonals of its three faces, you need to apply the formula:
volume = 1/8 × √(a² - b² + c²)(a² + b² - c²)(-a² + b² + c²),
where a, b, and c are the diagonals you’re given. This formula can be easily derived by using the Pythagorean theorem.
Solved Examples based on Volume of a rectangular prism
Problem 1: Find the height of a rectangular prism if its volume is 90 cubic inches and its base area is 15 square inches.
Solution:
Given data,
The volume of a rectangular prism = 90 cu. in
Base area = 15 sq. in
We know that,
The volume of rectangular prism formula = Base area × Height of the prism
⇒ 90 = 15 × h
⇒ h = 90/15 = 6 inches.
Hence, the height of the given prism is 6 inches.
Problem 2: Determine the volume of the rectangular prism if its base length is 10 cm, the base width is 6 cm and the height of the prism is 15 cm.
Solution:
Given data,
The height of the rectangular prism (h) = 15 cm
Base length (l) = 10 cm
Base width (w) = 6 cm
We know that,
The volume of rectangular prism formula (V) = Base area × Height of the prism
Base area = l × w
= 10 × 6 = 60 sq. cm.
V = 60 × 15 = 900 cu. cm
Hence, the volume of the rectangular prism is 900 cu. cm.
Problem 3: What is the base width of a rectangular prism if its volume is 2100 cu. cm and its height and base lengths are 25 cm and 12 cm, respectively?
Solution:
Given data,
The volume of a rectangular prism = 2100 cu. cm
The height of the rectangular prism (h) = 25 cm
Base length (l) = 12 cm
We know that,
The volume of rectangular prism formula (V) = Base area × Height of the prism
Base area = l × w
⇒ V = l × w × h
⇒ 2100= 12 × w × 25
⇒ 300w = 2100
⇒ w = 2100/300 = 7 cm
Hence, the base width of a rectangular prism is 7 cm.
Problem 4: What is the volume of a rectangular prism whose height is 20 units and whose base area is 120 square units?
Solution:
Given data,
The height of the rectangular prism (h) = 20 units
Base area = 120 square units
We know that,
The volume of rectangular prism formula (V) = Base area × Height of the prism
V = 120 × 20 = 2400 cubic units.
Hence, the volume of a rectangular prism is 2400 cubic units.
Problem 5: What is the base length of a rectangular prism if its volume is 150 cu. cm and its height and base widths are 10 cm and 3 cm, respectively?
Solution:
Given data,
The volume of a rectangular prism = 150 cu. cm
The height of the rectangular prism (h) = 10 cm
Base width (w) = 3 cm
We know that,
The volume of rectangular prism formula (V) = Base area × Height of the prism
Base area = l × w
⇒ V = l × w × h
⇒ 150 = l × 3 × 10
⇒ 30l = 150
⇒ l = 150/30 = 5 cm
Hence, the base length of a rectangular prism is 5 cm.
Problem 6: What is the volume of a rectangular prism whose height is 20 units and whose base length and width are 15 units and 12 units, respectively?
Solution:
Given data,
The height of the rectangular prism (h) = 20 units
Base length (l) = 15 units
Base width (w) = 12 units
We know that,
The volume of a rectangular prism = l × w × h cubic units
V = 15 × 12 × 20 = 3600 cubic units.
Hence, the volume of a rectangular prism is 3600 cubic units.
Problem 7: Determine the volume of a rectangular prism if its height is 10 cm and its base length and width are 8 cm and 6 cm, respectively.
Solution:
Given data,
The height of the rectangular prism (h) = 10 cm
Base length (l) = 8 cm
Base width (w) = 6 cm
We know that,
The volume of a rectangular prism = l × w × h cubic units
V = 8 × 6 × 10 = 480 cu. cm
Hence, the volume of a rectangular prism is 480 cu. cm.
FAQs based on Rectangular Prism
Question 1: What is the Volume of a Rectangular Prism?
Answer:
Volume of rectangular prism is the amount of substance that it can hold or it is the space occupied by it in 3-D space. So,volume of rectangular prism is calculated by multiplying its area of base with its height. Formula for finding the volume of a rectangular prism is,
Volume (V) = height of the prism × base area.
It is calculated in cubic units such as cm3, m3, in3, etc.
Question 2: What changes occur to the Volume of the Rectangular Prism if its height is doubled?
Answer:
Volume of a rectangular prism is the product of its three dimensions, i.e,
volume = length × width × height.
If the height of rectangular prism is doubled, its volume will be V2 = l × w × (2h) = 2 × lwh = 2 × V1. So, it is safe to say that the volume of the rectangular prism gets doubled when its height is doubled.
Question 3: What happens to the Volume of the Rectangular Prism if its height is halved?
Answer:
Volume of a rectangular prism is the product of its three dimensions, i.e,
volume = length × width × height.
If the height of rectangular prism is halved, its volume will be V2 = l × w × (h/2) = (lwh)/2 = V1/2. So, it is safe to say that the volume of the rectangular prism gets halved when its height is halved.
Question 4: What Happens to the Volume of Rectangular Prism if the Length, Width, and Height of Prism are Doubled?
Answer:
Volume of a rectangular prism is the product of its three dimensions, i.e,
volume = length × width × height.
If the length, breadth,and height of rectangular prism are doubled, its volume will be V2 = (2l) × (2w) × (2h) = 8 × lwh = 8 × V1. Hence, volume of the rectangular prism gets eight times when its all three dimension are doubled.
Formulas for a rectangular prism:
- Volume of Rectangular Prism:
- V = lwh
- Surface Area of Rectangular Prism:
- S = 2(lw + lh + wh)
- Space Diagonal of Rectangular Prism: (similar to the distance between 2 points)
- d = √(l2 + w2 + h2)
A cube is a special case where l = w = h. So you can find the volume of a cube or surface area of a cube by setting these values equal to each other.
Calculations for a rectangular prism:
1. Given the length, width and height find the volume, surface area and diagonal of a rectangular prism
- h, l and w are known; find V, S and d
- V = lwh
- S = 2(lw + lh + wh)
- d = √(l2 + w2 + h2)
2. Given the surface area, length and width find the height, volume and diagonal of a rectangular prism
- S, l and w are known; find h, V and d
- h = (S – 2lw) / (2l + 2w)
- V = lwh
- d = √(l2 + w2 + h2)
3. Given the volume, length and width find the height, surface area, and diagonal of a rectangular prism
- V, l and w are known; find h, S and d
- h = V / lw
- S = 2(lw + lh + wh)
- d = √(l2 + w2 + h2)
4. Given the diagonal, length and width find the height, volume and surface area of a rectangular prism
- d, l and w are known; find h, V and S
- h = √(d2 – l2 – w2)
- V = lwh
- S = 2(lw + lh + wh)
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