19.4 Surface area and volume of prisms and cylinders — Learn

Prerequisites: calculate areas of 2D shapes and multiply lengths; use circle formulae.

Objectives: calculate volumes and surface areas of prisms and cylinders.

Prisms and cylinders

A prism has a repeated end shape, called its cross-section, along its length. Its volume is $Ah$, where $A$ is cross-section area and $h$ is prism length. For a right prism, surface area is $2A+Ph$, where $P$ is cross-section perimeter. A cylinder is a right prism with circular ends. Its volume is $\pi r^2h$. The curved surface unwraps to a rectangle with sides $2\pi r$ and $h$, so its curved area is $2\pi rh$. Add two circular ends for total surface area.

Example 1. A triangular prism has cross-section area $15\text{ cm}^2$ and length 8 cm.

Multiply the cross-section area by the length: $15\times8=120$.

Its volume is $120\text{ cm}^3$.

Example 2. A cuboid is 5 cm long, 4 cm wide and 3 cm high. Work out its volume.

Multiply the base area by the height: $5\times4\times3=60$.

So its volume is $60\text{ cm}^3$.

Example 3. A triangular prism has cross-section area $6\text{ cm}^2$, cross-section perimeter 12 cm and length 5 cm. Work out its surface area.

Add the two triangular ends and three rectangular faces: $2\times6+12\times5=72$.

So its surface area is $72\text{ cm}^2$.

Example 4. A cylinder has radius 3 cm and height 7 cm. Find its volume and total surface area.

Use $V=\pi r^2h$: $V=\pi\times3^2\times7=63\pi$.

Add the curved area and two ends: $2\pi\times3\times7+2\pi\times3^2=60\pi$.

So volume is $63\pi\text{ cm}^3$ and surface area is $60\pi\text{ cm}^2$.

Watch out. Surface area uses square units and volume uses cubic units. A missing face often means you counted an internal join.

Progress check

A cylindrical kerb around the pond has radius $0.5$ m and length $6$ m. Work out its volume in terms of $\pi$. Then name its cross-section and the dimensions of its curved surface.

You should now be able to

• find volume by multiplying cross-section area by prism length;
• calculate cylinder volume using $\pi r^2h$;
• find exposed face areas for total surface area;
• distinguish square units from cubic units.