In geometry, a prism is a three-dimensional shaped object that has two identical polygons facing each other and is laterally connected by rectangular or parallelogram faces. These identical polygons are called the bases of the prism and can be of any shape, such as triangles, squares, rectangles, or any n-sided polygon. A prism is a significant member of the polyhedron family. Depending upon the type of the polygonal base, a prism is classified into two types: regular and irregular prisms. And depending upon the alignment of the bases, there are two types of prisms: right prisms and oblique prisms. Furthermore, based on the shape of the polygonal base a prism is classified into different types: triangular prism, rectangular prism, square prism, pentagonal prism, hexagonal prism, etc.
Rectangular Prism
A rectangular prism is a three-dimensional shape consisting of six rectangular flat faces. It is a prism with two rectangular bases and four lateral rectangular faces, twelve sides, and eight vertices. According to mathematical studies, a cuboid is any polyhedron that resembles a rectangular prism. In a rectangular prism, every angle is a right angle. It is also known as a rectangular hexahedron, a right rectangular prism, and a rectangular parallelepiped.
Volume of a Rectangular Prism
The volume of a rectangular prism is the total space enclosed within the rectangular prism. It is usually represented by the letter “V” and is measured in terms of cm3, m3, in3, etc. A rectangular prism’s volume is determined by multiplying its base area by its height.
The formula for the volume of a rectangular prism is given as follows:
Volume of a rectangular prism = Base area × Height of the prism
Since the base of the prism is a rectangle, its area is the product of its length and width. Let “h” be the height of the prism, “l” be the base length, and “b” be the base width.
Now, the formula for the volume of a rectangular prism is given as follows:
Volume of a rectangular prism = l × w × h cubic units
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.
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