Volume of a Prism

Consider the right prism as shown in the figure below. There are as many cubic units in each layer parallel to the base ABCD as there are square units in the area of the base. Also, there are as many layers of cubic units in the prism as there are linear units in the altitudes h. We can, therefore, find the total number of cubic units in the prism by multiplying the area of the base by the altitude.


vol-prism

\begin{gathered} V = {\text{Area }} \times {\text{ Height}} \\ \therefore V = A \times h \\ \end{gathered}


Volume of the prism  = Area of the base  \times Height of the prism

 

Rule: The volume of the prism equals its base times its altitude.

 

Example:

The base of a right prism is an equilateral triangle with a side of 4 cm and its height is 25 cm. Find its volume.

Solution:
Volume

V = {\text{Area }} \times {\text{ Height}}

Since the base is an equilateral triangle:
\therefore {\text{Area}} = \frac{{\sqrt 3 }}{4}{\left( {{\text{one side}}} \right)^2} = \frac{{\sqrt 3 }}{4}{\left( 4 \right)^2} = 4\sqrt 3 = 6.92 cm
h = 25 cm
\therefore V = {\text{Area }} \times {\text{ Height}} = 6.25 \times 25 = 173.20 cubic cm

 

Example:

The sides of a triangular prism are 17 cm, 25 cm and 28 cm respectively. The volume of the prism is 4200 cubic cm. What is its height?

 

Solution:

Now, A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)}

\because s = \frac{{a + b + c}}{2} = \frac{{17 + 25 + 28}}{2} = \frac{{70}}{2} = 35


\therefore A = \sqrt {35\left( {35 - 17} \right)\left( {35 - 25} \right)\left( {35 - 28} \right)} = \sqrt {35 \times 18 \times 10 \times 7} = 210 square cm

\begin{gathered} \therefore V = A \times h \Rightarrow 4200 = 210 \times h \\ \Rightarrow h = 20cm \\ \end{gathered}