Year 11 General Shape And Measurement

Similarity And Scaling

10 practice questions 2 video lessons Theory + worked examples

Theory

Two figures are similar if one is an enlargement of the other — same shape, different size. This page covers scale factors, the rules for scaling lengths, areas and volumes, map scale problems, and similar-triangle shadow problems.

Two figures are similar when one is an enlargement of the other: they have the same shape but possibly different sizes. All matching angles are equal, and all matching sides are in the same ratio. That ratio is called the scale factor, or the linear ratio.

If the linear ratio between two similar figures is $a : b$ , then sides and perimeters scale as $a : b$ , areas scale as $a^{2} : b^{2}$ , and volumes scale as $a^{3} : b^{3}$ . Doubling every length therefore multiplies area by $4$ and volume by $8$ .

A scale drawing or map uses a ratio of the form $1 : k$ , meaning $1$ unit on the drawing represents $k$ units in real life. So a map scale of $1 : 25 000$ means $1$ cm on the map is $25 000$ cm $= 250$ m on the ground.

Two similar rectangles. Linear ratio

2 : 3

gives area ratio

2^{2} : 3^{2} = 4 : 9

A pole and a tree with shadows from the same sun form similar right triangles (AA).

For two similar figures (or solids) with matching sides in the linear ratio $a : b$ :

Quantity	Ratio
Lengths (sides, perimeters)	$a : b$
Areas (and surface areas)	$a^{2} : b^{2}$
Volumes (and capacities)	$a^{3} : b^{3}$

Map and scale-drawing rules (scale $1 : k$ ):

actual length = drawing length \times k

actual = drawing \times k

drawing length = \frac{actual length}{k}

drawing = \frac{actual}{k}

Going backwards. If you know an area ratio and want the linear ratio, take the square root. From a volume ratio, take the cube root. For example: area ratio $4 : 9$ gives linear ratio $2 : 3$ ; volume ratio $8 : 125$ gives linear ratio $2 : 5$ .

How to solve any similarity/scaling problem

Identify the linear ratio $a : b$ between the two figures (it may be given directly, or you may need to derive it from a known length, area, or volume).
Decide which power of the ratio applies: lengths use $a : b$ , areas use $a^{2} : b^{2}$ , volumes use $a^{3} : b^{3}$ .
Set up a proportion (a fraction-equals-fraction) using matching quantities, then solve for the unknown.

Map shortcut. For a scale $1 : k$ , just multiply (drawing → actual) or divide (actual → drawing) by $k$ . Convert units at the end so the answer is in the form the question asks for.

Example 1 — Map Scale

A map has a scale of

1 : 50 000

. Two towns are

8

cm apart on the map. Find the actual distance, in km.

Solution

Multiply the map distance by the scale denominator, then convert.

actual	$=$	$8 \times 50 000$
actual	$=$	$400 000 cm$
actual	$=$	$4 000 m$
actual	$=$	$4 km$

actual = 4 km

Example 2 — Shadow Problem

1.6

m post casts a

0.8

m shadow. At the same time, a tree casts a

15

m shadow. Find the height of the tree.

Solution

The two right triangles are similar (same sun angle, both vertical).

$\frac{tree}{15}$	$=$	$\frac{1.6}{0.8}$
$\frac{tree}{15}$	$=$	$2$
tree	$=$	$30 m$

tree = 30 m

Example 3 — Area Scales as k Squared

A small rectangle has dimensions

3

\times

5

cm. A similar larger rectangle has matching sides

3

times longer. Find the area of the larger rectangle.

Solution

Find the small area, then multiply by $3^{2}$ .

$A_{small}$	$=$	$3 \times 5 = 15 {cm}^{2}$
$A_{large}$	$=$	$15 \times 3^{2}$
$A_{large}$	$=$	$15 \times 9$
$A_{large}$	$=$	$135 {cm}^{2}$

A_{large} = 135

Example 4 — Volume Scales as k Cubed

Two similar containers have a linear ratio of

2 : 5

. The smaller holds

80

L. Find the capacity of the larger.

Solution

Volume ratio is $2^{3} : 5^{3} = 8 : 125$ .

$\frac{80}{V_{large}}$	$=$	$\frac{8}{125}$
$V_{large}$	$=$	$\frac{80 \times 125}{8}$
$V_{large}$	$=$	$1 250 L$

V_{large} = 1250

Common pitfalls

Ratios out of order. Always pair smaller with smaller (or larger with larger) on both sides of the equation. Don't mix

\frac{small}{large}

with

\frac{large}{small}

in the same equation.

Wrong power for area or volume. Area scales by the square of the linear ratio (

k^{2}

); volume scales by the cube (

k^{3}

). A common slip is using

k

instead.

Square-rooting a ratio.

\sqrt{4 : 9} = 2 : 3

, not

2 : 4.5

. Take the root of each number separately.

Mixed units on maps. Always convert to the unit the question asks for. A

1 : 50 000

map gives

8 cm \to 400 000 cm

; convert to km (divide by

100 000

) only at the end.

Frequently asked questions

What is a scale factor?

A scale factor is the number you multiply by to go from one similar figure to another. If two similar figures have matching sides in the ratio 2 to 3, then the scale factor from the smaller to the larger is 3 over 2, or 1.5. All matching lengths are scaled by the same factor.

If I double all the sides of a shape, what happens to the area and volume?

Area is multiplied by 2 squared which is 4, and volume is multiplied by 2 cubed which is 8. The general rule for similar figures with linear ratio a to b: lengths scale as a to b, areas scale as a squared to b squared, and volumes scale as a cubed to b cubed.

What does a map scale of 1:25000 mean?

It means 1 unit on the map represents 25000 units in the real world. So 1 cm on the map is 25000 cm, which equals 250 metres on the ground.

How do I work out the actual distance from a map?

Multiply the distance measured on the map by the scale denominator. If the scale is 1 to k, then actual distance equals map distance times k. Be careful with units: 1 cm on a 1 to 50000 map is 50000 cm equals 500 m equals 0.5 km.

Are two similar triangles always the same size?

No. Similar means same shape (same three angles) but possibly different sizes. Matching sides are in the same ratio, called the scale factor. Triangles that are also the same size are called congruent, which is a special case of similar with scale factor 1.

Why does volume scale faster than length?

Because there are three independent dimensions in volume (length, width, height) and each is multiplied by the same scale factor. So if you scale all three lengths by k, volume picks up a factor of k times k times k, which is k cubed.