Hard GCSE Similar lengths, areas and volumes Questions

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.

Write down the ratio of the two known volumes.

The volumes are in the ratio $1 : 54$, so the volume scale factor is $\frac{54}{1} = 54$. Because volumes scale by $k^3$, this fraction is $k^3$ — it is NOT $k$.

Take the cube root to get the length scale factor.

Undoing $k^3$ needs a cube root, so $k = \sqrt[3]{54} = 3\sqrt[3]{2}$. This is the step candidates forget: a volume ratio is not a length ratio.

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 $54$ is $k^3$, not $k$. Using it as the length scale factor — forgetting the cube root — is the commonest error here.

Check the length scale factor by raising it back to the right power.

Raising $3\sqrt[3]{2}$ to the power $3$ gives back $54$, the volume ratio the question states, so the scale factor is right.

Check the size of the scale factor.

$54$ lies between $27$ and $64$, so the scale factor lies between $3$ and $4$. The exact value $3\sqrt[3]{2}$ must be left as it is — rounding it early would spoil the answer.

Check the scale factor is bigger than one.

The second shape is the bigger one, so its lengths must be more than the first shape, and $3\sqrt[3]{2} > 1$ confirms it.

Note that a scale factor has no units.

A scale factor is a ratio of two lengths, so the units cancel: it is just a number, with no centimetres attached.

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.

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 how the same scale factor drives every other measurement.

Once $k$ is known, areas scale by $9\sqrt[3]{4}$ and volumes scale by $54$.

Note the reverse scale factor.

Going the other way uses the reciprocal of the scale factor. The two shapes are similar either way round.

Check that no rounding has crept in.

The scale factor $3\sqrt[3]{2}$ is exact. A rounded scale factor, cubed, throws a volume out badly, so it is kept exact all the way.

Choose the correct option.

The volume scale factor is $54$, so the length scale factor is its cube root, $3\sqrt[3]{2}$.

Write the percentage increase as a length scale factor.

An increase of $50\%$ multiplies every length by $1 + \frac{50}{100} = 1.5$, so the length scale factor is $1.5$.

Work out the area scale factor.

Areas scale by $k^2$, so the area is multiplied by $\left(1.5\right)^2 = 2.25$.

Turn the scale factor into a percentage increase.

A multiplier of $2.25$ means the area becomes $225\%$ of what it was, which is an INCREASE of $125\%$. Subtracting the $1$ is what turns a multiplier into an increase.

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 area does NOT go up by $50\%$. Only the lengths do; the area goes up by $125\%$.

Check the answer with a concrete example.

Taking a length of $10$ to start with, it becomes $15$, so the area goes from $100$ to $225$ — an increase of $125$ out of $100$, which is $125\%$.

Check the percentage increase is the bigger one.

Scaling up a area magnifies the change, so the percentage increase in the area must be larger than the $50\%$ increase in the lengths.

Write the multiplier as a fraction.

Keeping the multiplier as an exact fraction, rather than a rounded decimal, is what keeps the final percentage exact.

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 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 the original size does not matter.

The starting size cancels out, so the percentage increase is the same whatever the shape started at. That is why no measurements were needed.

Note the multiplier form of the answer.

Multiplying by $2.25$ and increasing by $125\%$ are two ways of saying the same thing.

Note what a decrease would look like.

Had the lengths been decreased, the scale factor would be less than $1$, and the area would shrink even faster than the lengths.

State the percentage increase in the area.

The area increases by $125\%$.

Write down the ratio of the two known areas.

The areas are in the ratio $25 : 9$, so the area scale factor is $\frac{9}{25} = \frac{9}{25}$. Because areas scale by $k^2$, this fraction is $k^2$ — it is NOT $k$.

Take the square root to get the length scale factor.

Undoing $k^2$ needs a square root, so $k = \sqrt{\frac{9}{25}} = 0.6$. This is the step candidates forget: a area ratio is not a length ratio.

Work out the volume scale factor.

Volumes scale by $k^3$, so the volume scale factor is $\left(0.6\right)^3 = 0.216$.

Multiply the volume of the larger solid by that scale factor.

Multiplying the volume of the larger solid, $250$, by the scale factor $0.216$ gives the volume of the smaller solid as $54$.

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 $k \times k = k^2$.

Check the units of the answer.

The volume of the smaller solid is measured in cubic centimetres, so the answer is $54\text{ cm}^3$.

Check the answer points the right way.

The volume of the smaller solid must be smaller than the one you were given, and $54 < 250$, so it is.

Note the standard mistake in this type of question.

Multiplying by the LENGTH scale factor $0.6$ instead of by $k^3$ is the commonest error here. Volumes need $k^3$.

Check the length scale factor by raising it back to the right power.

Raising $0.6$ to the power $2$ gives back $\frac{9}{25}$, the area ratio the question states, so the scale factor is right.

Check the answer by working backwards.

Dividing the answer by the scale factor gives back $250$, the value the question started from, so the multiplication is right.

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 $k^3$.

Write the length, area and volume ratios side by side.

The lengths are in the ratio $5 : 3$, so the areas are in the ratio $25 : 9$ and the volumes are in the ratio $125 : 27$. Squaring and cubing a ratio means squaring and cubing BOTH parts.

Note the alternative way of setting the work out.

Some candidates prefer to write the two quantities as a fraction and set it equal to $k^3$, then solve. It is the same calculation, arranged differently.

State the volume of the smaller solid.

The volume of the smaller solid is $54$ cubic centimetres.

Write down the ratio of the two known volumes.

The volumes are in the ratio $1 : 8$, so the volume scale factor is $\frac{8}{1} = 8$. Because volumes scale by $k^3$, this fraction is $k^3$ — it is NOT $k$.

Take the cube root to get the length scale factor.

Undoing $k^3$ needs a cube root, so $k = \sqrt[3]{8} = 2$. This is the step candidates forget: a volume ratio is not a length ratio.

Work out the area scale factor.

Areas scale by $k^2$, so the area scale factor is $\left(2\right)^2 = 4$.

Multiply the surface area of the smaller solid by that scale factor.

Multiplying the surface area of the smaller solid, $15$, by the scale factor $4$ gives the surface area of the larger solid as $60$.

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 $k \times k = k^2$.

Check the units of the answer.

The surface area of the larger solid is measured in square centimetres, so the answer is $60\text{ cm}^2$.

Check the answer points the right way.

The surface area of the larger solid must be bigger than the one you were given, and $60 > 15$, so it is.

Note the standard mistake in this type of question.

Multiplying by the LENGTH scale factor $2$ instead of by $k^2$ is the commonest error here. Areas need $k^2$.

Check the length scale factor by raising it back to the right power.

Raising $2$ to the power $3$ gives back $8$, the volume ratio the question states, so the scale factor is right.

Check the answer by working backwards.

Dividing the answer by the scale factor gives back $15$, the value the question started from, so the multiplication is right.

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 $k^3$.

Write the length, area and volume ratios side by side.

The lengths are in the ratio $1 : 2$, so the areas are in the ratio $1 : 4$ and the volumes are in the ratio $1 : 8$. Squaring and cubing a ratio means squaring and cubing BOTH parts.

Note the alternative way of setting the work out.

Some candidates prefer to write the two quantities as a fraction and set it equal to $k^2$, then solve. It is the same calculation, arranged differently.

State the surface area of the larger solid.

The surface area of the larger solid is $60$ square centimetres.