Volume & Surface Area

Spec reference: Geometry and measures - Mensuration and calculation
Key idea: Calculate volume and surface area of 3D shapes.

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Volume formulae

Shape	Volume
Cuboid	$V=lwh$
Prism	$V=\text{cross-section area}\times \text{length}$
Cylinder	$V=\pi r^{2}h$
Cone	$V=\frac{1}{3}\pi r^{2}h$
Sphere	$V=\frac{4}{3}\pi r^3$
Pyramid	$V=\frac{1}{3}\times \text{base area}\times h$

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Surface area

Shape	Surface area
Cuboid	$SA=2(lw+lh+wh)$
Cylinder	$SA=2\pi r^2+2\pi rh$
Sphere	$SA=4\pi r^2$
Cone	$SA=\pi r^2+\pi rl$ (where $l$ = slant height)

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Worked examples

Example

Find the volume of a cylinder with radius 4 cm and height 10 cm.

$$V=\pi \times 4^2\times 10=160\pi \approx \boxed{{\displaystyle 503{\text{ cm}}^3}}$$

Example

Find the surface area of a cuboid $5\times 3\times 2$ cm.

$$SA=2(5\times 3+5\times 2+3\times 2)=2(15+10+6)=2\times 31=\boxed{{\displaystyle 62{\text{ cm}}^2}}$$

Example

Find the volume of a cone with radius 6 cm and height 9 cm.

$$V=\frac{1}{3}\times \pi \times 36\times 9=108\pi \approx \boxed{{\displaystyle 339{\text{ cm}}^3}}$$

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Prisms

A prism has the same cross-section throughout its length.

$V=\text{area of cross-section}\times \text{length}$

Example

A triangular prism has a right-angled triangle cross-section with base 6 cm and height 4 cm. Length = 10 cm.

Area of triangle $=\frac{1}{2}\times 6\times 4=12$ cm²

$V=12\times 10=\boxed{{\displaystyle 120{\text{ cm}}^3}}$

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Exam tips

Watch out for

• Cone and pyramid volume: don't forget the $\frac{1}{3}$
• Cylinder surface area has TWO circular ends: $2\pi r^2+2\pi rh$
• Give exact answers in terms of $\pi$ unless told to use a decimal

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Test Yourself

Question 1 of 5

Volume of cylinder, $r=5$, $h=8$ (in terms of $\pi$)?