GCSE Volume and surface area of prisms Practice Questions

Recall the rule for the volume of a cuboid.

A cuboid is a prism with a rectangular cross-section, so its volume is the three edge lengths multiplied together.

Substitute the three edge lengths.

The edges are $3$ cm, $4$ cm and $5$ cm.

State the volume of the cuboid.

The volume of the cuboid is $60$ $\text{cm}^3$.

Recall the rule for the volume of a prism.

A prism has the same cross-section all along its length, so its volume is the area of that cross-section multiplied by the length.

Rule out the other expressions.

Adding $A$ and $l$ makes no sense because they have different units, and dividing them would give a length, not a volume. The halving and doubling options would count the wrong amount of the solid: a prism is exactly one cross-section repeated along its length.

Select the correct expression.

The volume of a prism is its cross-sectional area multiplied by its length, so $V = A l$.

Start from the length conversion.

There are $100$ centimetres in a metre.

Cube the length conversion.

A cubic metre is a cube of side $1$ m, which is a cube of side $100$ cm, so its volume is $100$ cubed cubic centimetres.

Carry out the multiplication.

So one cubic metre is $1000000$ cubic centimetres.

Rule out the length and area factors.

The factor $100$ converts lengths and $100^2 = 10000$ converts areas; only $100^3$ converts volumes.

Rule out the remaining options.

Neither $1000$ nor $100000$ is $100$ cubed, so neither can be right.

Select the correct number.

There are $1000000$ cubic centimetres in a cubic metre.

Recall the volume of a cylinder.

A cylinder is a prism, so its volume is the area of its circular cross-section multiplied by its height.

Work out the area of the circular cross-section.

The circle has area $36\pi$ $\text{cm}^2$.

Form an equation for the height.

The volume is $288\pi$ $\text{cm}^3$, so the cross-sectional area multiplied by the height must equal it.

Cancel pi and divide.

The factor of $\pi$ cancels, leaving a height of $8$ cm.

Check by working the volume forwards.

Substituting the height back gives $288\pi$ $\text{cm}^3$, the volume in the question.

Test one of the other heights.

A height of $48$ cm would give the wrong volume, so that statement is false.

Test another of the other heights.

A height of $24$ cm is wrong too.

Note the slip that produces the largest wrong option.

Dividing by the radius instead of by the radius squared gives $48$, one of the tempting wrong answers.

Check the units of the height.

A volume divided by an area leaves a length, so the height is in centimetres.

Select the correct statement.

The height of the cylinder is $8$ cm.

Realise that the amount of paper is the surface area.

The paper covers the outside of the box, so the area of paper needed is the total surface area of the cuboid.

Work out the three different face areas.

The three kinds of face have areas $240$, $200$ and $120$ square centimetres.

Add the three face areas and double.

Each face appears twice, so the box needs $1120$ $\text{cm}^2$ of paper.

Multiply the area by the cost of one square centimetre.

Each square centimetre costs $2$ pence.

Carry out the multiplication.

The paper costs $2240$ pence.

State the units of the answer.

Two lengths in centimetres are multiplied together, so an area is measured in square centimetres, $\text{cm}^2$.

Check the area against the net.

Adding the six rectangles of the net one at a time gives the same $1120$ $\text{cm}^2$.

Check the cost by dividing back.

Dividing the cost by $2$ pence per square centimetre returns the area $1120$ $\text{cm}^2$.

Convert the cost into pounds for a sanity check.

That is about 22.4 pounds, a believable price for a sheet of paper this size.

Note the common slip of using the volume.

The volume of the box is $2400$ $\text{cm}^3$, but paper covers a SURFACE, so it is the area, not the volume, that is needed.

Note that no paper is wasted in this model.

The calculation assumes the paper is cut to fit exactly, with no overlaps and no waste, which is what an exam question of this kind intends.

Identify the largest face.

The largest face is $240$ $\text{cm}^2$, so no single sheet smaller than that could cover it.

Check the surface area is bigger than the largest face.

The whole surface must be far bigger than any one face, and it is.

State the cost per face as a check.

Every square centimetre of the $1120$ needed costs $2$ pence, giving $2240$ pence in all.

State the total cost of the paper.

The paper costs $2240$ pence.