Volume of prisms

A prism is a solid with a uniform cross-section. This means that no matter where it is sliced along its length, the cross section is the same size and shape (congruent).

Diagram of a cylinder split into slices

A well-known example of a prism is a cylinder and you can see from the image above that the front face (cross-section) is the same size of circle no matter where you slice it.

\[V = Ah\]

Question

Calculate the volume of the prism.

Hexagonal prism with a length of 12cm and a face area of 54cm≤

\[V = Ah\]

\[= 54 \times 12\]

\[= 648c{m^3}\]

Question

Calculate the volume of the prism.

Triangular prism with dimensions of 12cm x 6cm x 4cm

This shape is a triangular prism so the area of the cross-section is the area of a triangle.

Area of the triangle:

\[A = \frac{1}{2}bh\]

\[= \frac{1}{2} \times 6 \times 4\]

\[= 12c{m^2}\]

Volume of the prism:

\[V = Ah\]

\[= 12 \times 12\]

\[= 144c{m^3}\]

Volume of a cylinder

The formula for the volume of a cylinder (circular prism) is derived from the volume of a prism, where \(r\) is the radius and \(h\) is the height/length.

\[V = Ah\]

Since the area of a circle \(= \pi {r^2}\), then the formula for the volume of a cylinder is:

\[V = \pi {r^2}h\]

Question

Calculate the volume of the cylinder shown.

Give your answer correct to 1 significant figure.

Diagram of a cylinder with a height of 10cm and a radius of 4cm

\[V = \pi {r^2}h\]

\[= \pi \times {4^2} \times 10\]

\[= 502.654...\]

\[= 500c{m^3}\,(to\,1\,s.f.)\]