Sphere, Cone & Pyramid (SQA National 5 Maths): Revision Note

From your National 4 Maths course, you should also be familiar with calculating the volumes of cubes, cuboids, prisms and cylinders.

The volume formulae for spheres, cones and pyramids are all included on the Formulae List in the exam paper.

Be careful with sphere and cone questions.

• The question may give you the diameter instead of the radius

• Remember to halve the diameter to find the radius before substituting into the formula

A sphere has a diameter of 9 centimetres.

Calculate the volume of the sphere, and round the final answer to 3 significant figures.

Answer:

The sphere's diameter is 9 cm

• This means the radius, $r$, is $\frac{9}{2}=$4.5 cm

Use the formula for volume of a sphere, $V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\times \pi \times 4.5^3$

Use your calculator to evaluate that

$=381.703507...$

Round to 3 significant figures, as required, and don't forget to include units

382 cm³ (3 s.f.)

The diagram below shows a cone with diameter 20 centimetres and height 90 centimetres.

Without using a calculator, calculate the volume of the cone.

Take $\mathit{\pi }\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{14}$.
 

Answer:

The cone's base diameter is 20 cm

• This means the radius, $r$, is $\frac{20}{2}=$10 cm

Use the formula for volume of a cone, $V=\frac{1}{3}\pi r^{2}h$

• Use $\pi =3.14$, as the question says

$\begin{array}{rcl}V& =& \frac{1}{3}\times 3.14\times {10}^2\times 90\\ & =& \frac{1}{3}\times 90\times {10}^2\times 3.14\end{array}$

$\frac{1}{3}\times 90=30$ and ${10}^2=100$

$\begin{array}{rcl}& =& 30\times 100\times 3.14\\ & =& 3000\times 3.14\\ & =& 3\times 3.14\times 1000\\ & =& 9.42\times 1000\\ & =& 9420\end{array}$

Don't forget to include units in your final answer

9420 cm³