Defining the Term Volume
All three-dimensional shapes occupy space known as their volume. We also use the term “capacity” to express the maximum amount of liquid that an object can hold or contain. The volume of a three-dimensional shape can also be explained as the number of unit cubes that fit inside the shape. Hence, volume can be expressed in cubic units such as cubic meters, cubic inches, cubic centimeters and so on. Cups, pints, quarts and gallons, are other commonly used units to measure volume.
Defining the Term Prisms
Since the characteristics of three-dimensional shapes vary from one shape to the other we use different formulas to find their volume. Our focus will be on the formulas used to find the volumes of a group of three-dimensional shapes known as prisms.
Prisms are three-dimensional solid shapes that have two identical faces on opposite sides. These sides are similar and parallel to each other. In the next section we will learn the formulas used to calculate the volume of rectangular prisms and cubes.
List of formulas
Since rectangular prisms and cubes have slightly different characteristics, we use different formulas to find their volumes. But the formula used to calculate the volume of a cube is derived from the formula used to calculate the volume of a rectangular prism.
- Volume of a rectangular prism = l x w x h
Where, l, w, and h, are the length, width and height of the prism.
- Volume of a cube = \(a^3\)
Where, a is the length of the side of a cube.
A rectangular prism, or a cuboid, is a three-dimensional shape with a pair of parallel and congruent bases. A cuboid has six rectangular faces. We need to know the length (l), width (w) and height (h) of a cuboid to find its volume. So, the volume of a cuboid is the product of its length, width and height.
The formula used to calculate the volume of a cube is derived from the formula for the volume of a cuboid . That is because a cube is a special type of cuboid in which its length, width and height are of the same measure. So, the formula l x w x h can be rewritten as, \(a^3\), where, l = w = h = a.
Solved Examples
Example 1: Find the area of the rectangular prism.
Solution:
For the given rectangular prism,
l = 10 inches
w = 5 inches
h = 4 inches
Volume of a rectangular prism = l x w x h
= 10 x 5 x 4 [Substitute values]
= 200 [Multiply]
So, the volume of the cuboid is 200 cubic inches.
Example 2: Find the volume of the cube.
Solution:
The edge length of the cube is 8 inches.
Volume of cube = \(a^3\)
= \(8^3\) [Substitute 8 for a]
= 512 [Cube of 8]
So, the volume of the cube is 512 cubic inches.
Example 3: A water tank is in the shape of a rectangular prism. If it is 3 feet long and 5 feet tall, and it can hold 60 cubic feet of water, find the width of the water tank.
Solution:
Length of the water tank = 3 feet
Width of the water tank = ?
Height of the water tank = 5 feet
The capacity of the water tank = 60 cubic feet
Volume of the tank = l x w x h
60 = 3 x w x 5 [Substitute values]
60 = 15 x w [Multiply]
w = 60 ÷ 15 [Divide both sides by 15]
w = 4
So, the water tank is 4 feet wide.
Example 4: A swimming pool is in the shape of a rectangular prism and its dimensions are as shown in the image. Find the amount of water it can hold when filled up.
Solution:
To find the amount of water that the swimming pool can hold, we need to find the volume of the pool.
The pool is in the shape of a rectangular prism.
So, volume of the swimming pool = l x w x h
Here, l = 30 feet
w = 15 feet
h = 6 feet
Volume = 30 x 15 x 6 [Substitute values]
= 2700 [Multiply]
Hence, the swimming pool can hold 2700 cubic feet of water when it is filled up.
Example 5: A cube of edge length 10 inches is kept on top of another cube of the same dimensions to form a rectangular prism as shown in the image. Calculate the volume of the rectangular prism.
Solution:
A cube of side 10 inches is kept on top of another cube of the same dimensions to form a rectangular prism.
Height of the rectangular prism = 10 in + 10 in = 20 in
Width of the rectangular prism = 10 in
Length of the rectangular prism = 10 in
Volume of the rectangular prism = l x w x h
= 20 x 10 x 10 [Substitute values]
= 4000 [Multiply]
Therefore, the volume of the rectangular prism is 4000 cubic inches.
Cubes are three-dimensional solids that have six identical square faces. A cube can be considered as a special type of prism. This means any side of a cube can be chosen as the base of the prism.
Cubes are three-dimensional shapes that have identical square faces, and cuboids are three-dimensional shapes that have rectangular faces. By definition, a square is also a rectangle. Hence, we can consider a cube as a special case of cuboids in which the length, width and height are equal.
All the sides of a cube are made up of squares and have the same dimensions. On the other hand, a rectangular prism or a cuboid is a shape made up of rectangular faces.
The volume of the cube formula is derived from the volume of the cuboid formula. Since the length, width and height of a cube are the same, the formula l x w x h can be simplified, and written as a3.
A prism is a three-dimensional figure that has two identical bases on opposite ends. Whereas a pyramid is a three-dimensional figure that has only one base. The vertices of the base of a pyramid are connected to a point known as the apex of the pyramid.