A rectangular prism is a three-dimensional geometric shape with two congruent and parallel rectangular bases and four rectangular faces connecting them. This type of prism is also known as a cuboid. It is a member of the polyhedron family and is characterized by its six rectangular faces, twelve edges, and eight vertices. Common examples include books, cereal boxes, containers, and rooms.
Properties of Rectangular Prisms:
- Faces: 6 rectangular faces
- Edges: 12 edges
- Vertices: 8 vertices
There are two main types of rectangular prisms:
- Right Rectangular Prism: The bases are aligned perpendicularly to the height.
- Oblique Rectangular Prism: The bases are not aligned perpendicularly to the height.
Volume of a Rectangular Prism
The quantity of tri space contained in a surface is expressed as a scalar quantity called volume. For example, the amount of space occupied or contained by material or 3D objects. The SI-derived unit, the cubic meter, is frequently used to quantify volume numerically.
The entire space inside a rectangular prism is measured by the volume of the prism. Consider a water-filled rectangular container. In this situation, the box's capacity is equal to the entire amount of water it can store.
The formula for the volume of a rectangular prism is equal to the product of its base area and its height.
The volume of a Rectangular Prism (V) = Base Area × Height of the Prism
Formula
V = l × b × h
Where l, b, and h are the length, breadth, and height of the rectangular prism respectively.
Solved Examples on Volume of a Rectangular Prism
Example 1: Find the volume of a rectangular prism whose length, breadth, and height are 12, 15, and 8 cm.
Solution:
Given: l = 12 cm
b = 15 cm
h = 8 cm
Volume = l × b × h
= 12 × 15 × 8
= 1440 cm3
Example 2: Find the base area of a rectangular prism whose volume is 40 cm3 and height is 4 cm.
Solution:
Given: V = 40 cm3
h = 4 cm
Since, V = l × b × h or,
V = Base Area × Height
40 cm3 = Base Area × 4 cm
⇒ Base Area = 40/4
= 10 cm2
Example 3: Find the volume of a rectangular prism if its base area is 50 cm2 and height is 12 cm.
Solution:
Given: Base area = l × b = 50 cm2
h = 12 cm
Volume = l × b × h
= 50 × 12
= 600 cm3
Example 4: Find the height of a rectangular prism given its volume is 600 cm3 and the base area is 50 cm2.
Solution:
Given: V = 600 cm3
Base area = l × b = 50 cm2
Volume = l × b × h
600 = 50 × h
h = 600/50
= 12 cm
Example 5: Determine the volume of a rectangular prism if its height is 15 inches and the length and breadth of its base are 11 inches and 6 inches, respectively.
Solution:
Given data,
l = 11 inches
b = 6 inches
h = 15 inches
We know that,
The volume of a Rectangular Prism = (l × b × h) cubic units
= 12 × 11 × 6 = 792 cubic inches
Hence, the volume of the given prism is 792 cu. in.
Example 6: Find the volume of a rectangular prism whose length, breadth, and height are 10, 9, and 8 cm.
Solution:
Given: l = 10 cm
b = 9 cm
h = 8 cm
Volume = l × b × h
= 10 × 9 × 8
= 720 cm3
Example 7: Determine the volume of a rectangular prism if its height is 10 cm and the length and breadth of its base are 8 cm and 5 cm, respectively.
Solution:
Given data,
l = 8 cm
b = 5 cm
h = 10 cm
We know that,
The volume of a Rectangular Prism = (l × b × h) cubic units
= 10 × 8 × 5 = 400 cu. cm
Hence, the volume of the given prism is 400 cu. cm.
Practice Problems - Volume of a Rectangular Prism Formula
- Calculate the volume of a rectangular prism with dimensions 7 cm (length), 5 cm (breadth), and 9 cm (height).
- If a rectangular prism has a volume of 200 cm³ and a base area of 40 cm², what is its height?
- Find the base area of a rectangular prism with a volume of 150 cm³ and a height of 5 cm.
- Determine the height of a rectangular prism with a volume of 600 cm³ and a base area of 75 cm².
- Calculate the volume of a rectangular prism with a length of 4 inches, breadth of 3 inches, and height of 7 inches.
- A rectangular prism has a length of 12 cm, a height of 10 cm, and a volume of 720 cm³. What is its breadth?
- Find the volume of a rectangular prism with a height of 20 cm and a base area of 80 cm².
- If a rectangular prism's length is twice its breadth and the height is 10 cm, find the volume if the base area is 120 cm².
- A rectangular prism has dimensions 6 cm × 7 cm × 8 cm. Calculate its volume.
- Determine the height of a rectangular prism with a volume of 500 cm³ and a base area of 50 cm².
Summary
A rectangular prism is a fundamental 3D shape in geometry characterized by its six rectangular faces, twelve edges, and eight vertices. Understanding the properties and volume calculations of rectangular prisms is essential in various practical applications, from packaging to architectural design. The volume formula V=l×b×h is crucial for determining the space contained within the prism, aiding in solving real-world problems.