# 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.

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

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

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