Volume of 3-D Figures - Prisms Formulas | List of Volumes of 3-D Figures - Prisms Formulas You Should Know - BYJUS

# Volumes of 3-D Figures - Prisms Formulas

Volume of a three dimensional object is the amount of space it encloses. A prism is a solid three-dimensional shape with two identical faces and other faces that resemble parallelograms. Here we will learn about the formula used for calculating the volume of a prism....Read MoreRead Less

### 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.