Calculating the Volume of a Prism
A prism is a polyhedron with flat faces and parallel bases. Prisms are characterized by identical ends and the same cross-section throughout their entire length. The base of the prism is in the shape of a polygon and that is how they are classified. For instance, a prism with a triangular base is known as a triangular prism. Similarly, there are rectangular prisms, pentagonal prisms, hexagonal prisms and so on.
Volume of a Prism Formulas
The general formula to find the volume of a prism is given below.
Volume of a Prism, V = Base Area, B × Length
Where, ‘B’ is the area of the base of the prism.
Cubic units are the measurement units used to represent the volume.
How to use the Volume of a Prism Formulas?
The volume of a prism is the product of its base area and length. For a triangular prism the base area will be obtained by using the area of a triangle formula. Similarly for a rectangular prism the base area will be obtained by using the area of the rectangle formula.
The table below shows the volume of triangular and rectangular prisms.
Volume of a triangular prism:
A prism with three rectangular faces and two triangular bases is known as a triangular prism. Because the cross-section of a triangular prism is a triangle, the volume of a triangular prism can be calculated using the following formula:
The volume of a Triangular Prism = \((\frac{1}{2})\) l x b x h cubic units.
Where,
l = length of a triangular prism
b = Base length of a triangular prism
h = height of a triangular prism
Volume of a rectangular prism:
Four rectangular faces and two parallel rectangular bases make up a rectangular prism. A rectangular prism’s cross-section is known to be a rectangle. A “Cuboid” is another name for the rectangular prism. As a result, the formula for calculating the volume of a rectangular prism is:
The volume of a Rectangular Prism = l x b x h cubic units.
Where
l = Base width of a rectangular prism
b = Base length of a rectangular prism
h = height of a rectangular prism
Solved Examples
Example 1: A brick making machine makes rectangular prism-shaped bricks that are 7 inches long, 6 inches wide and 3 inches tall. What volume of concrete is needed to make a single brick?
Solution:
Volume of concrete needed will be the volume of the brick.
Length of the brick = 7 inches.
Width of the brick = 6 inches.
Height of the brick = 3 inches.
The brick is in the shape of a rectangular prism.
So,
Volume of brick = length x width x height [Volume of rectangular prism formula]
= 7 x 6 x 3 [Substitute values]
= 126 cu. inches [Multiply]
Therefore, the volume of concrete needed to make a single brick is 126 cubic inches.
Example 2: Emily has moved into a new house that has an attic as shown here. What is the volume of the attic?
Solution:
The attic is of the shape of a triangular prism.
Triangular base area, B
= \(\frac{1}{2}\) x 10 x 12 = 60 sq. ft
Volume of the attic, V = B x Length [Volume of triangular prism formula]
= 60 x 16 [Substitute values]
= 960 cu. ft [Multiply]
Therefore, The volume of the attic is 960 cubic feet.
Example 3: If the volume of the prism shown is 684 cubic centimeters, what will its length be?
Solution:
Given data,
Base of the triangle, b = 9.5 cm.
Height of the triangle, h = 4.8 cm.
Therefore, area of the triangle = \(\frac{1}{2}\) x 9.5 x 4.8
= \(\frac{1}{2}\) x 45.6
= 22.8 sq.cm
Volume of the prism = area of the triangular base x length
684 = 22.8 x length [Substitute values]
length = \(\frac{684}{22.8}\) = 30 [Simplify]
Therefore, the length is 30 centimeters.
Example 4: The lateral surface area of the triangular prism shown in the image is 1056 square centimeters. Find its volume.
Solution:
Side length of the triangular base of the prism, s = 16 cm
Perimeter of the base of the prism
= s + s + s = 16 + 16 + 16 = 48 cm
Lateral surface area of the triangular prism
= perimeter of the base of the prism x length of the prism
1056 = 48 x length of the prism [Substitute values]
length of the prism = \(\frac{1056}{48}\) = 22 cm [Simplify]
Volume of the triangular prism
= area of the triangular base x length of the prism
Area of the triangular base = \(\frac{\sqrt{3}}{4}~s^2\) [Area of equilateral triangle formula]
= \(\frac{\sqrt{3}}{4}~(16)^2\) [Substitute values]
= \(\frac{1.732}{4}\) x 256 [Simplify]
= 1.732 x 64 [Further simplify]
= 110.848 sq.cm [Multiply]
So,
Volume of the triangular prism = 110.848 x 22
= 2438.656 [Multiply]
Hence, volume of the triangular prism is 2438.656 cu.cm.
Example 5: What is the volume of this doll house?
solution:
The doll house consists of a triangular prism mounted on a rectangular prism so we will calculate the volume of the two shapes separately.
For the triangular prism,
Volume of a Triangular Prism = (\(\frac{1}{2}\)) l x b x h
= (\(\frac{1}{2}\)) 40 x 10 x 30 [Substitute values]
= 6000 [Multiply]
The volume of the triangular prism part of the doll house is 6000 cu.cm.
For the rectangular prism,
Volume of a Rectangular Prism = l x b x h
= 40 x 10 x 15 [Substitute values]
= 6000 [Multiply]
The volume of the rectangular prism part is 6000 cu.cm.
The volume of the model doll house will be the sum of volume of the 2 prisms, that is,
= 6000 + 6000 = 12000 [Add]
Hence, the volume of the doll house is 12000 cu.cm.
Prisms are classified on the basis of the shape of their base and the volume of a prism is the function of its base area. Hence, for each type of prism, the shape of the base changes which in turn changes the volume of the prism.
If the base area and height of the prism are doubled, the volume of the prism will quadruple as base ‘B’ is substituted by ‘2B’ and length ‘L’ is substituted by ‘2L’.
As a result V = 2B x 2L = 4(BL), which is four times the prism’s original volume.
Taking the product of the base area and the prism’s length yields the formula for the volume of a prism. The volume of a prism is calculated using the formula V = B x L; where ‘V’ denotes the volume, ‘B’ the prism’s base area and ‘L’ the prism’s length.