Volume of Prism
- Prism
- Volume of Prism
- Types of Prism and their Volumes
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Prism
A prism is a three-dimensional polyhedral shape containing surface area and volume. There are different types of prisms, and each prism has a unique base. On this page, we will discuss the volume of a prism.
Volume of Prism
When an object occupies any space, the space it occupies is called the volume of the object (prism). We can find a prism's volume by multiplying its base area by height.
Where, `V =` Volume of the Prism
`B =` Base area
`H =` Height of the Prism
Types of Prism and Their Volume
This discussion will explore the different types of prism and the volumes of these types.
`1`. Triangle-shaped prism: It is a type of prism that consists of three rectangular faces and two triangular bases. It can also be called a triangular prism. Due to its triangle base, the prism’s volume can be calculated as follows:
Triangular Prism’s volume `=` Triangle’s area `xx` Height
Or, `V = B\times H`
Here is an image of a triangle-shaped prism.
`2`. Square Prism: It has `6` faces, including its base and height. In a square-shaped prism, both the height and the base sides are equal. It is also known as square prism. Since its base is equal to its other faces, the square-shaped prism’s volume can be determined as follows:
Square Prism’s volume `=` Square-shaped Base area `\times` Height of the prism
Or, `V = B\times H`
Here is an image of a square prism.
`3`. Rectangle-shaped prism: It has four rectangle-shaped faces and two parallel rectangle-shaped bases. It is also known as rectangular prism. Due to its rectangle-shaped base, the volume can be determined as follows:
Rectangular prism’s volume `=` Area of the rectangle base `xx` height of the prism
Or, `V = B\times H`
Here is the image:
`4`. Pentagonal Prism: It has five rectangle-shaped faces and two parallel pentagonal bases. It is also known as the pentagonal prism. Due to its pentagon-shaped base, the volume can be determined as follows:
Pentagonal prism’s volume `=` Pentagonal base’s area `xx` Height of the prism
Or, `V = B\times H`
Here is the image for more clarity.
`5`. Hexagonal Prism:
It has six rectangle-shaped faces and two parallel hexagon-shaped bases. It is known as hexagonal prism. Due to its hexagon-shaped base, the volume can be calculated as follows:
Hexagonal prism’s volume `=` Hexagonal base’s area `xx` Height of the prism
Or, `V = B\times H`
Here is an image for more clarity.
`6`. Octagon-shaped Prism:
It has eight rectangle-shaped faces and two parallel octagon-shaped bases. It is known as an octagonal prism. Due to its octagon-shaped base, the volume can be calculated as follows:
Octagonal prism’s volume `=` Octagonal base’s area `xx` Height of the prism
Or, `V = B\times H`
Here is an image:
Solved Examples
Example `1`: Determine the rectangular prism’s volume, where the base area is `15 ` square units and the height is `4` units.
Solution:
Rectangular prism's volume `=` Area of the rectangle base `xx` height of the prism
Volume `= 15` square units `xx` `4` units
Volume `= 60` cubic units
Example `2`: What is prism’s height, whose base area is `6` square centimeters and volume is `42` cubic centimeters?
Solution:
Given, the volume of the prism `= 42` cubic centimeters
Prism’s base area `= 6` square centimeters
Formula: Volume `=` Base area `xx` Height
`42` cubic centimeters `= 6` square centimeters `xx` Height
Height `= 42/6`
Height `= 7` centimeters
Thus, the height of the given prism is `7` centimeters.
Alternative method: `V = B\times H`, `42 = 6\times H`, `H = 42/6, H = 7` centimeters
Example `3`: Find the base area of a prism whose volume is `169` cubic units and the height is `13` units.
Solution:
Given, Volume `= 169` cubic units
Height `= 13` units
Formula: `V = B\times H`
`169 = B\times 13`
`B = 169/13`
`B = 13` square units.
Example `4`: Shelby received a rectangular prism as a birthday gift from her sister. Her sister told her that the base area is `36` cubic centimeters and the height is `9` centimeters. Now, she wants to calculate the volume of that prism. Can you help her find the volume of the prism?
Solution:
As we already know:
Base area `(B) = 36` square centimeters
Height `(H) = 9` centimeters
Now, she wants to determine the rectangular prism’s volume.
Formula: `V = B\times H`
`V = 36\times 9`
`V = 324` cubic centimeters
Thus, the rectangular prism’s volume is `324` cubic centimeters.
Example `5`: Determine a rectangular prism’s volume whose length of the base is `10` cm, the breadth of the base is `5` cm and the height of the prism is `110` cm.
Solution:
Given prism's height `= 110` cm
Length of the base `= 10` cm
Breadth of the base `= 5` cm
Since the base is rectangle-shaped,
Area of the rectangular base = length `xx` breadth
`= 10` cm `xx` `5` cm
`= 50` square centimeters
Therefore, the base area of the prism `(B) = 50` square centimeters
and, the height of the prism `(H) = 110` cm
Using the formula of the volume, `V = B\times H`
`V = 50\times 110`
`V = 5500` cubic centimeters
So, the rectangular prism’s volume is `5500` cubic centimeters.
Practice Problems
Q`1`. Determine a rectangular prism’s volume whose base area is `200` square units and the height is `80` units.
- `16000` cubic units
- `15000` cubic units
- `16500` cubic units
- `15600` cubic units
Answer: a
Q`2`. What is the prism’s height whose volume is `500` cubic centimeters and the base area is `50 ` square centimeters?
- `51` centimeters
- `10` centimeters
- `19` centimeters
- `3` centimeters
Answer: b
Q3. What is the base area of the triangle-shaped prism whose height is 11 square centimeters and volume is `1331` cubic centimeters?
- `120` square centimeters
- `110` square centimeters
- `121` square centimeters
- `210` square centimeters
Answer: c
Q4. The volume of a prism depends on-
- Number of faces
- Number of edges
- The base area and the height
- Number of vertices
Answer: c
Q5. What is the formula for a prism’s volume?
- `B = H\times V`
- `H = V\times B`
- `V = H\times E`
- `V = H\times B`
Answer: d
Frequently Asked Questions
Q`1`. What is a prism?
Answer: A prism is a three-dimensional geometric figure with two identical bases connected by rectangular or parallelogram faces.
Q`2`. What are the different types of prism?
Answer: There are different types of prism:
- Triangle-shaped Prism
- Square-shaped Prism
- Rectangle-shaped Prism
- Pentagon-shaped Prism
- Hexagon-shaped Prism
Q`3`. How do you classify prisms based on the shape of their bases?
Answer: Prisms are classified by the shape of their bases. For example, rectangular prisms have rectangular bases and triangular prisms have triangular bases.
Q`4`. What is the formula for calculating the volume of a prism?
Answer: The volume (\(V\)) of a prism is calculated using the formula \(V = \text{Base Area} \times \text{Height}\), where the base area is the area of one of the polygonal bases.
Q`5`. How can you determine if a prism is a right prism or an oblique prism?
Answer: In a right prism, the lateral faces are perpendicular to the base, forming right angles. In an oblique prism, the lateral faces are not perpendicular to the base.