Volume of Prism: Formulas | Types | Real-life Application

A prism is a polyhedron in which all faces are flat, and the bases are parallel to each other. It is a solid object with flat faces, identical ends, the same cross section, along with its length.

The volume of a prism is defined as the total shape occupied by a three-dimensional shape. It is the capacity of the prism. Mathematically, it is defined as the product of the area of the base and the length.

Thus, as each prism is a three-dimensional shape, the volume of a prism also lies in a three-dimensional plane. The unit of volume of a prism is given as cubic meters, cubic centimeters, cubic inches, cubic feet, etc.

Mathematically, the formula is:

Volume of Prism = Base Area × Height

Volume Formulas Table

The table below shows the volume formulas for different types of prisms:

Notation:

a = apothem length, b = base and h = height

Steps to Calculate the Volume of Prism

To calculate the volume of a prism, you need to know the area of its base and its height (or length). The formula for the volume of a prism is: Volume = Base Area × Height.

Step 1: Find the Base Area
The base of a prism can have different shapes (e.g., triangle, rectangle, etc.).

• For example:
  If the base is a rectangle, the area is calculated as: Base Area = Length × Width
  If the base is a triangle, the area is: Base Area = 1/2 × Base × Height of the Triangle

Step 2: Find the Height
The height of the prism is the perpendicular distance between the two bases.

Step 3: Multiply the Base Area by the Height
Once you know the base area and the height, multiply them to find the volume.

Different Types of Prisms and Their Volume Formulas

Now let us discuss the volume of different prism formulas such as the volume of a triangular prism, rectangular prism, pentagonal prism, and so on.

Triangular prism

Triangular prism is a prism that has three rectangular faces and two triangular faces as bases. Since the cross section of the triangular prism is a triangle.

Volume of Triangular Prism = (1/2)abh cubic meter

Rectangular Prism

Rectangular prism is a prism that has four rectangular faces and two parallel rectangular bases. The cross section of rectangular prism is rectangle. The rectangular prism is also know as "Cuboid".

Volume of Rectangular Prism = l.b.h cubic meter

Pentagonal Prism

Pentagonal prism has five rectangular faces and two parallel pentagonal bases . Since the base area of pentagonal prism is (5/2) ab.

Volume of Pentagonal Prism = (5/2) a.b.h

Hexagonal Prism

Hexagonal Prism is a prism with six rectangular faces and two parallel hexagonal bases. The base area of the hexagonal prism is 3ab.

Volume of Hexagonal Prism = 3abh cubic meter

Square Prism

Square Prism is a special type of cuboid (rectangular prism) in which the bases are square.

Volume of Square Prism = a² x l

Octagonal Prism

An octagonal prism is a 3D shape with two octagonal bases and eight rectangular sides. Think of it like a can of soda, but instead of a circle, the top and bottom are octagons, and the sides are rectangles.

Volume of Octagonal Prism = 2(2 + √ 2)a² x h

Trapezoidal Prism

Trapezoidal Prism is a geometric solid characterized by its unique trapezoidal base and parallel rectangular faces.

Volume of Trapezoidal Prism = 1/2( a + b) x h x l

Heptagonal Prism

A heptagonal prism is a 3D shape solid consisting of two identical heptagonal bases joined by seven rectangular faces.

Volume of Heptagonal Prism = 7/2 x a² x cot(π/7) x h

Real-Life Application of Prism:

• Triangular prism - It is found in roof structure and bridge stability.
• Rectangular prism - It is used in storage containers and furniture design.
• Pentagonal prism - It is seen in pencil cases and decorative items.
• Hexagonal prism - It is used in bolts, nuts, and pencils for easy handling.
• Octagonal prism - It is used in a stop sign and traffic board.
• Square prism - Most houses, bricks, and rooms are shaped like a square prism.
• Heptagonal prism - Certain coins (like UK 50p and 20p) resemble a heptagonal prism.
• Trapezoidal prism - It is used to design bridge support.

Related Reads:

• Geometric Shapes in Maths
• Real Life Applications of Prism
• Volume of a Cone

Solved Questions on the Volume of Prism

Question 1: Find the height of the prism if the volume of the prism is 729 cubic units and the base area is 27 square units.

Solution:

Here,
Volume of the prism = 729 cubic meter
Base area = 27 square units

Let the height of prism be h
Substituting the values in the volume of prism formula
V = B × H = 729 cubic meter
27B = 729 cubic meter
H = 27 units

Therefore ,the height of prism is 27 uints

Question 2: A triangular prism has a the volume of 360 cm³.The triangular base has a base length of the prism of 12 cm and base height of 6cmthe . Find the height of prism.

Solution :

Here,
Volume of triangular prism = 360 cm³
Base of prism = 12 cm
Base height = 6 cm

Volume of triangular prism = (1/2) × base length × base height × height
360 = (1/2) × 12 × 6 × h
360 = 36h
h = 10 cm

Therefore , The height of prism is 10 cm.

Question 3: A square prism has a square base of 12 cm and height of 6 cm. Find the volume of prism .

Solution:

Here,
base = 12 cm
height = 6 cm

Volume of square prism = a² × l
V = (12)² × 6
V = 144 × 6
V = 864 cm ³

Therefore , The volume of square prism is 864 cm³

Question 4: Find the volume of a rectangular prism whose width is 7 cm, height is 6 cm, and length is 16 cm.

Solution:

Here,
width = 7 cm
height = 6 cm
length = 16 cm

Volume of rectangular prism = l × b × h
V = 16 × 6 × 7
V = 960 cm ³

Therefore, volume of rectangular prism is 960 cm ³

Question 5: Find the volume of a pentagonal prism whose base is 6 cm, apothem is 10 cm and height is 10 cm.

Solution:

Here,
a = 10 cm
b = 6 cm
h = 10 cm

Volume of pentagonal prism = (5/2)a.b.h
V = 5/2 × 10 ×6 ×10
V = 5/2 × 576
V = 1440 cm ³

Question 6: Find the volume of a trapezoidal prism whose base edges are 8 cm, and 12 cm, vertical height is 5 cm, and length is 15 cm.

Solution:

Here,
a = 8 cm
b = 12 cm
h = 5 cm
l = 15 cm

Volume of trapezoidal prism = 1/2(a + b) × h × l
V = 1/2 (8 + 12) × 5 × 15
V = 1/2(20) × 75
V = 10 × 75
V = 750 cm ³

Volume of trapezoidal prism is 750 cm³

Unsolved Questions on the Volume of Prism

Question 1: Find the volume of the prism whose base area is 27 cm ² and height is 15 cm.

Question 2: Find the volume of a rectangular prism whose width is 7 cm, height is 12 cm, and length is 16 cm.

Question 3: Find the volume of a pentagonal prism whose base is 10 cm, apothem is 8 cm and height is 12.5 cm.

Question 4: Find the volume of a hexagonal prism whose base is 6 cm, the apothem is 5.2 cm and the height is 10 cm.

Question 5: A square prism has a volume of 750 cm³. The side length of the square base is 5 cm. Find the height of the prism.

Question 6: An octagonal prism has a regular octagonal base with a side length of 6 cm and an apothem of 8 cm. The height of the prism is 10 cm. Find the volume of a prism.

Answer Sheet

1) 405 cm³
2) 1344 cm³
3) 2500 cm³
4) 936 cm³
5) 30 cm
6) 2304 cm³