GCSE Similar lengths, areas and volumes Practice Questions

Work out the length scale factor.

The corresponding lengths are in the ratio $1 : 3$. Going from the smaller shape to the larger shape, every length is multiplied by $\frac{3}{1} = 3$, and that is the LENGTH scale factor $k$.

Work out the area scale factor.

Areas scale by $k^2$, so the area scale factor is $\left(3\right)^2 = 9$.

State the area of the larger shape.

The area of the larger shape is $45$ square centimetres.

Work out the length scale factor.

The corresponding lengths are in the ratio $1 : 3$. Going from the smaller shape to the larger shape, every length is multiplied by $\frac{3}{1} = 3$, and that is the LENGTH scale factor $k$.

Work out the area scale factor.

Areas scale by $k^2$, so the area scale factor is $\left(3\right)^2 = 9$.

Choose the correct calculation.

Areas scale by $k^2$, so the area of the larger shape is $12 \times 3^2 = 108$.

Write down the ratio of the lengths that the question gives.

The two solids are mathematically similar, and their lengths are in the ratio $1 : 3$.

Raise each part of the length ratio to the power of 3.

Volumes scale by $k^3$, so BOTH parts of the ratio are raised to the power of $3$, giving $1 : 27$.

State how lengths, areas and volumes scale.

If two shapes are mathematically similar with length scale factor $k$, then every area is multiplied by $k^2$ and every volume is multiplied by $k^3$.

Recall why volumes scale by the cube of the scale factor.

A volume is built from three lengths multiplied together, so enlarging every length by $k$ multiplies the volume by $k^3$.

Note the standard mistake in this type of question.

The ratio $1 : 3$ is a ratio of lengths. Quoting it as the ratio of the volumes, without changing it, is the commonest error here.

Choose the correct statement.

Every length is multiplied by $3$, so the volume is multiplied by $3^3 = 27$.

Write down the ratio of the areas that the question gives.

The two shapes are mathematically similar, and their areas are in the ratio $16 : 81$.

Take the square root of each part to get the ratio of the lengths.

Areas scale by $k^2$, so undoing that needs the square root of EACH part: the lengths are in the ratio $4 : 9$.

State how lengths, areas and volumes scale.

If two shapes are mathematically similar with length scale factor $k$, then every area is multiplied by $k^2$ and every volume is multiplied by $k^3$.

Recall why areas scale by the square of the scale factor.

An area is built from two lengths multiplied together. Enlarging both lengths by $k$ multiplies the area by $k \times k = k^2$.

Note the standard mistake in this type of question.

The ratio $16 : 81$ is a ratio of areas. Quoting it as the ratio of the lengths, without changing it, is the commonest error here.

Check that both parts of the ratio were treated the same way.

A ratio is only unchanged if both parts are treated identically, so both parts are raised to the power of $1$ — not just one of them.

Check the ratio is in its simplest form.

$4$ and $9$ have no common factor other than $1$, so the ratio $4 : 9$ cannot be cancelled any further.

Write the ratio as a scale factor.

The length scale factor is $\frac{9}{4}$, and the length scale factor is that raised to the power of $1$, which is $\frac{9}{4}$.

Check the answer with actual lengths.

A shape with a length of $4$ scaling up to one with a length of $9$ gives a length ratio of $4 : 9$, which cancels to $4 : 9$.

Choose the correct option.

The areas are in the ratio $16 : 81$, so the lengths are in the ratio $4 : 9$ — the square root of each part.

Link mass to volume.

Both solids are made from the same material, so they have the same density. Mass is density times volume, so the masses are in exactly the same ratio as the volumes — and volumes scale by $k^3$.

Write down the ratio of the lengths that the question gives.

The two spheres are mathematically similar, and their lengths are in the ratio $1 : 2$.

Raise each part of the length ratio to the power of 3.

Volumes scale by $k^3$, so BOTH parts of the ratio are raised to the power of $3$, giving $1 : 8$.

State how lengths, areas and volumes scale.

If two shapes are mathematically similar with length scale factor $k$, then every area is multiplied by $k^2$ and every volume is multiplied by $k^3$.

Recall why volumes scale by the cube of the scale factor.

A volume is built from three lengths multiplied together, so enlarging every length by $k$ multiplies the volume by $k^3$.

Note the standard mistake in this type of question.

The ratio $1 : 2$ is a ratio of lengths. Quoting it as the ratio of the volumes, without changing it, is the commonest error here.

Check that both parts of the ratio were treated the same way.

A ratio is only unchanged if both parts are treated identically, so both parts are raised to the power of $3$ — not just one of them.

Check the ratio is in its simplest form.

$1$ and $8$ have no common factor other than $1$, so the ratio $1 : 8$ cannot be cancelled any further.

Write the ratio as a scale factor.

The length scale factor is $\frac{2}{1}$, and the volume scale factor is that raised to the power of $3$, which is $\frac{8}{1}$.

Check the answer with actual lengths.

A sphere with a length of $1$ scaling up to one with a length of $2$ gives a volume ratio of $1 : 8$, which cancels to $1 : 8$.

Recall why areas scale by the square of the scale factor.

An area is built from two lengths multiplied together. Enlarging both lengths by $k$ multiplies the area by $k \times k = k^2$.

Note what mathematically similar means.

Mathematically similar shapes have equal angles and all their corresponding lengths in the same ratio. That single ratio is what drives the area and volume scale factors.

Note that a ratio has no units.

Both parts of the ratio are measured in the same units, so the units cancel and the ratio is just a pair of numbers.

Note the reverse ratio.

Reading the two spheres the other way round reverses the ratio to $8 : 1$. The order the question asks for is the order to give.

Choose the correct statement.

Mass is proportional to volume for the same material, and volume scales by $k^3$, so the mass is $2^3 = 8$ times as big.