4.11 Similarity

Learning objectives

4.11A Geometrical properties of similar figuresunderstand and use the geometrical properties that similar figures have corresponding lengths in the same ratio but corresponding angles remain unchanged 4.11B Maps and scale drawingsUse and interpret maps and scale drawings.

Similar figures have equal corresponding angles and all corresponding lengths in one constant ratio. The orientation may change, so correspondence must be established before calculating.

Step	Action
1	match equal angles or distinctive vertices
2	write corresponding sides in the same order
3	find linear scale factor $k=\frac{\text{new}}{\text{original}}$
4	multiply every original length by $k$

A valid scale factor gives the same ratio for every corresponding side. Angles remain unchanged and are never multiplied by $k$ .

Do not pair sides merely because they occupy the same place on the page; rotated or reflected similar figures can reverse the visual order.

A scale links a measured drawing length to a real length. For '1 cm represents 80 km', a drawing measurement of $d$ cm represents $80d$ km.

Direction	Operation
drawing to real	measure, then multiply by scale value
real to drawing	convert units, then divide by scale value
ratio scale $1:n$	1 drawing unit equals $n$ real units

Measure between the specified points with the same ruler convention used by the scale, keep units explicit, and allow for stated measurement tolerance before comparing routes or distances.

A straight-line map distance is not automatically the distance travelled along roads, and centimetres cannot be combined directly with kilometres.

If corresponding lengths have scale factor $k$ , corresponding areas have scale factor $k^2$ . Conversely, an area scale factor $a$ gives linear scale factor $\sqrt a$ .

Known	Required factor
length factor $k$	area factor $k^2$
area factor $a$	length factor $\sqrt a$
area ratio $A_2:A_1$	length ratio $\sqrt{A_2/A_1}$

Keep the direction consistent—new divided by original—then apply the squared factor to every corresponding area, including curved surface area.

Doubling every length makes area four times as large, not twice as large; area is two-dimensional.

If corresponding lengths have scale factor $k$ , corresponding volumes have scale factor $k^3$ . Conversely, a volume scale factor $v$ gives linear scale factor $\sqrt[3]{v}$ .

Known	Required factor
length factor $k$	volume factor $k^3$
volume factor $v$	length factor $\sqrt[3]{v}$
volume ratio $V_2:V_1$	length ratio $\sqrt[3]{V_2/V_1}$

Match corresponding solids, write the factor direction, cube only the linear factor, and attach cubic units to the result.

A volume ratio is not squared: a solid has three scaled dimensions, so the linear factor is cubed.

For the same pair of similar figures, one linear factor $k$ controls all measures: lengths scale by $k$ , areas by $k^2$ , and volumes by $k^3$ .

From	To	Operation
area factor	length factor	square root
volume factor	length factor	cube root
area factor	volume factor	take square root, then cube
volume factor	area factor	take cube root, then square

Choose one direction and convert the given ratio back to the linear factor before moving to the required dimension. Apply totals or differences only after corresponding measures have been expressed consistently.

Do not apply an area ratio directly to a volume or vice versa; both must pass through the common linear scale factor.

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—Pearson Edexcel International GCSE Mathematics (Specification A) (4MA1)Read

4.11 Similarity

Learning objectives

Use corresponding features of similar figures

Use maps and scale drawings

Use area scale factors

Use volume scale factors

Connect lengths, areas and volumes of similar figures

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