When one shape is an enlargement of another by scale factor $k$, three different things change — and they change by three different amounts.
| Length factor $k$ | Area factor $k^2$ | Volume factor $k^3$ |
|---|---|---|
| $2$ | $4$ | $8$ |
| $3$ | $9$ | $27$ |
| $4$ | $16$ | $64$ |
| $5$ | $25$ | $125$ |
| $1.5$ | $2.25$ | $3.375$ |
| $0.5$ | $0.25$ | $0.125$ |
| $\tfrac{2}{3}$ | $\tfrac{4}{9}$ | $\tfrac{8}{27}$ |
- Find $k$ by dividing a length on the new shape by the matching length on the old one.
- Decide what you are scaling: a length, an area or a volume.
- Multiply by $k$, $k^2$ or $k^3$ as appropriate.
- Sense-check the direction — bigger shape should give bigger answers.
Two similar triangles have corresponding sides $5$ cm and $12$ cm. The smaller has area $20\text{ cm}^2$. Find the area of the larger.
Two similar jugs have heights $10$ cm and $15$ cm. The smaller holds $500$ ml. Find the capacity of the larger.
Two similar cuboids have corresponding edges $18$ cm and $12$ cm. The larger has volume $1080\text{ cm}^3$. Find the volume of the smaller.
From a volume ratio: $k = \sqrt[3]{\text{volume factor}}$
Two similar shapes have areas $32\text{ cm}^2$ and $200\text{ cm}^2$. A side of the smaller is $6$ cm. Find the matching side of the larger.
Two similar cones have volumes $54\text{ cm}^3$ and $250\text{ cm}^3$. The smaller has height $9$ cm. Find the height of the larger.
Two similar solids have volumes in the ratio $8 : 125$. Find the ratio of their surface areas.
Surface areas are in the ratio $4 : 25$.
| Lengths | Areas | Volumes |
|---|---|---|
| $1 : 2$ | $1 : 4$ | $1 : 8$ |
| $2 : 3$ | $4 : 9$ | $8 : 27$ |
| $3 : 5$ | $9 : 25$ | $27 : 125$ |
| $1 : 4$ | $1 : 16$ | $1 : 64$ |
Two similar solid statues are cast from the same bronze. Their heights are $20$ cm and $50$ cm. The smaller has mass $4$ kg. Find the mass of the larger.
Rectangle $A$ is $6$ cm by $9$ cm. Rectangle $B$ is $10$ cm by $15$ cm. Are they similar?
Check with areas: $54\text{ cm}^2$ and $150\text{ cm}^2$; $\dfrac{150}{54} = \dfrac{25}{9} = \left(\dfrac{5}{3}\right)^2$ ✓
The big three
Length $\times k$, area $\times k^2$, volume $\times k^3$.
Finding $k$
New length $\div$ old length, using corresponding sides.
From areas
$k = \sqrt{\text{area factor}}$.
From volumes
$k = \sqrt[3]{\text{volume factor}}$.
Ratios
Lengths $a:b$ → areas $a^2:b^2$ → volumes $a^3:b^3$.
Mass
Same material: mass scales like volume, $\times k^3$.
Congruence
The special case $k = 1$ — nothing changes.
Testing similarity
Corresponding sides must all be in the same ratio.
Keep fractions
Exact fractions make squares and cubes far tidier.
Two similar shapes have lengths in the ratio $4 : 7$. Write the ratio of (a) their areas, (b) their volumes.
▶ Show solution
(a) $4^2 : 7^2 = 16 : 49$
(b) $4^3 : 7^3 = 64 : 343$
A shape is enlarged by scale factor $5$. Its area was $12\text{ cm}^2$. Find the new area.
▶ Show solution
Area factor $= 5^2 = 25$
New area $= 12 \times 25 = 300\text{ cm}^2$
Two similar cylinders have radii $4$ cm and $10$ cm. The smaller has volume $96\text{ cm}^3$. Find the volume of the larger.
▶ Show solution
$k = \dfrac{10}{4} = 2.5$
Volume factor $= 2.5^3 = 15.625$
Volume $= 96 \times 15.625 = 1500\text{ cm}^3$
Two similar shapes have areas $45\text{ cm}^2$ and $80\text{ cm}^2$. A length on the smaller is $9$ cm. Find the matching length on the larger.
▶ Show solution
Area factor $= \dfrac{80}{45} = \dfrac{16}{9}$
$k = \sqrt{\dfrac{16}{9}} = \dfrac{4}{3}$
Length $= 9 \times \dfrac{4}{3} = 12$ cm
Two similar solids have volumes $128\text{ cm}^3$ and $432\text{ cm}^3$. Find the ratio of their surface areas.
▶ Show solution
Volume ratio $= 128 : 432$. Divide both by $16$: $\;8 : 27$.
$8 = 2^3$ and $27 = 3^3$, so the length ratio is $2 : 3$.
Surface area ratio $= 2^2 : 3^2 = \mathbf{4 : 9}$
Two similar bottles have heights $12$ cm and $18$ cm. The taller holds $810$ ml. Find the capacity of the shorter.
▶ Show solution
Going tall to short: $k = \dfrac{12}{18} = \dfrac{2}{3}$
Volume factor $= \left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27}$
Capacity $= 810 \times \dfrac{8}{27} = 240$ ml
Two similar solid spheres are made of the same material. Their radii are $3$ cm and $6$ cm. The smaller has mass $450$ g. Find the mass of the larger.
▶ Show solution
$k = \dfrac{6}{3} = 2$
Mass scales like volume: factor $= 2^3 = 8$
Mass $= 450 \times 8 = 3600$ g $= 3.6$ kg
Rectangle $P$ measures $8$ cm by $12$ cm. Rectangle $Q$ measures $12$ cm by $16$ cm. Determine whether they are similar, showing your working.
▶ Show solution
$P$: $\dfrac{8}{12} = \dfrac{2}{3} = 0.667$
$Q$: $\dfrac{12}{16} = \dfrac{3}{4} = 0.75$
The ratios are different, so the rectangles are not similar.
A model car is made to a scale of $1 : 24$. The real car has a boot with capacity $480$ litres and a windscreen of area $0.72\text{ m}^2$.
(a) Find the model boot's capacity in millilitres. (b) Find the model windscreen's area in cm².
▶ Show solution
$k = \dfrac{1}{24}$ going from real to model.
(a) Volume factor $= \dfrac{1}{24^3} = \dfrac{1}{13\,824}$
$480 \div 13\,824 = 0.03472$ litres $= 34.7$ ml (1 d.p.)
(b) Area factor $= \dfrac{1}{24^2} = \dfrac{1}{576}$
$0.72 \div 576 = 0.00125\text{ m}^2$
$0.00125 \times 10\,000 = 12.5\text{ cm}^2$
A cone of height $24$ cm is cut by a plane parallel to its base, $8$ cm from the apex, producing a small cone and a frustum. The full cone has volume $1728\text{ cm}^3$ and curved surface area $432\text{ cm}^2$.
(a) Find the volume of the small cone. (b) Find the volume of the frustum. (c) Find the curved surface area of the small cone. (d) Write the ratio of the volume of the small cone to the volume of the frustum.
▶ Show solution
The small cone is similar to the whole cone, with $k = \dfrac{8}{24} = \dfrac{1}{3}$.
(a) Volume factor $= \left(\dfrac{1}{3}\right)^3 = \dfrac{1}{27}$
Small cone $= 1728 \times \dfrac{1}{27} = \mathbf{64\text{ cm}^3}$
(b) Frustum $= 1728 - 64 = \mathbf{1664\text{ cm}^3}$
(c) Area factor $= \left(\dfrac{1}{3}\right)^2 = \dfrac{1}{9}$
Curved surface $= 432 \times \dfrac{1}{9} = \mathbf{48\text{ cm}^2}$
(d) $64 : 1664$. Divide both by $64$: $\;\mathbf{1 : 26}$