Key ideas
Similar shapes
One shape is an enlargement of the other.
The angles match, and every length is multiplied by the same number.
The scale factor
Scale factor = new lengthcorresponding old length
Bigger than 1: an enlargement. Less than 1: a reduction.
Matching sides
In triangles, matching sides are opposite equal angles.
With parallel lines, the small and large triangles share an angle.
Lengths, areas and volumes
Lengths× scale factor
Areas× (scale factor)²
Volumes× (scale factor)³
Working back: square root an area scale factor, or cube root a volume scale factor, to get the linear scale factor.
In the exam
Usually 3 marks: •1 a valid scale factor •2 use it correctly (scale, square or cube) •3 the answer.
Without a calculator, keep the scale factor as a fraction, like 37. A rounded decimal loses a mark.
Scale factor and lengths
The two photographs are similar. Calculate the height, h, of the larger photograph.(2 marks)
Work out the scale factor.Scale factor = 156 = 52✓1
In plain EnglishThe widths match: 6 cm on the small photo and 15 cm on the large one. We are going from small to large, so the large width goes on top. Simplify the fraction.
Multiply the matching length.h = 4 × 52h = 10 cm✓2
In plain EnglishEvery length is multiplied by the same scale factor, so the height 4 cm becomes 4 × 5/2 = 10 cm.
Check: the scale factor is bigger than 1 because the photo gets bigger, and 10 cm is bigger than 4 cm.
Similar triangles
The two triangles are similar. Matching angles are marked. Calculate x.(2 marks)
Match the sides.5 cm matches 7.5 cm, and 4 cm matches xScale factor = 7.55 = 32✓1
In plain EnglishCorresponding sides are opposite equal angles. The sides opposite the angle with one arc are 5 cm and 7.5 cm, so they match. The large triangle is turned round, so match by the angles, not the position.
Scale the matching side.x = 4 × 32x = 6 cm✓2
In plain EnglishThe side opposite the two-arc angle is 4 cm in the small triangle and x in the large one.
AB is parallel to DE. AE and BD cross at C. Calculate the length of CE.(3 marks)
Spot the similar triangles.Triangles ABC and EDC are similarAB matches ED, AC matches EC
In plain EnglishParallel lines make equal alternate angles (Z angles), and the angles at C are vertically opposite. So the triangles are similar. Side AC is opposite angle B, and its match EC is opposite angle D.
Work out the scale factor.Scale factor = 128 = 32✓1
In plain EnglishAB = 8 cm and DE = 12 cm are the matching parallel sides. We are going from the small triangle to the large one.
Scale the matching side.CE = 6 × 32✓2CE = 9 cm✓3
In plain EnglishAC = 6 cm matches CE.
Triangles inside triangles
Triangles ABC and ADE are similar. BC is parallel to DE. Calculate the length of BD.(3 marks)
Work out the scale factor.Scale factor = 410 = 25✓1
In plain EnglishBC = 4 cm matches DE = 10 cm. We know AD, the side of the large triangle, and want AB in the small one, so go from large to small: the small length goes on top.
Scale the matching side.AB = 15 × 25 = 6 cm✓2
In plain EnglishAD = 15 cm in the large triangle matches AB in the small triangle.
Take away to find BD.BD = 15 − 6BD = 9 cm✓3
In plain EnglishBD is the piece of AD that is not part of AB.
⚠ Careful: BD is not the side of either triangle. Find AB first, then take it away. A rounded decimal for the scale factor loses •2 without a calculator.
Similar areas
Two flags are similar. The small flag is 12 cm long with an area of 96 cm². The large flag is 18 cm long. Calculate its area.(3 marks)
Linear scale factor.Scale factor = 1812 = 32✓1
In plain EnglishUse the lengths, not the areas: going from the 12 cm flag to the 18 cm flag.
Square it for area.Area scale factor = 322 = 94✓2
In plain EnglishAreas grow by the scale factor squared, because an area is a length times a length.
Multiply the area.Area = 96 × 94Area = 216 cm²✓3
In plain English96 × 9/4: 96 ÷ 4 = 24, and 24 × 9 = 216.
⚠ A common mistake: 96 × 3/2 = 144 cm². That uses the linear scale factor for an area. Areas need the scale factor squared.
Similar volumes
Two bottles are similar. The small bottle is 20 cm tall and holds 400 ml. The large bottle is 30 cm tall. Calculate its volume.(3 marks)
Linear scale factor.Scale factor = 3020 = 32✓1
In plain EnglishCompare the heights: going from small to large.
Cube it for volume.Volume scale factor = 323 = 278✓2
In plain EnglishVolumes grow by the scale factor cubed: length × length × length.
Multiply the volume.Volume = 400 × 278Volume = 1350 ml✓3
In plain English400 ÷ 8 = 50, and 50 × 27 = 1350.
Working back
Two similar logos have areas of 20 cm² and 45 cm². The small logo is 6 cm wide. Calculate the width of the large logo.(3 marks)
Area scale factor.Area scale factor = 4520 = 94✓1
In plain EnglishThis time we know both areas, so start with the area scale factor: large area over small area.
Square root for the linear scale factor.Linear scale factor = 94 = 32✓2
In plain EnglishArea scale factor = (linear scale factor)², so undo the square with a square root. 9 = 3 and 4 = 2.
Scale the length.Width = 6 × 32Width = 9 cm✓3
In plain EnglishWidths are lengths, so use the linear scale factor.
Two similar jugs hold 1000 ml and 343 ml. The large jug is 30 cm tall. Calculate the height of the small jug.(3 marks)
Volume scale factor.Volume scale factor = 3431000✓1
In plain EnglishGoing from the large jug to the small jug, so the small volume goes on top.
Cube root for the linear scale factor.Linear scale factor = 3431000 = 710✓2
In plain English7 × 7 × 7 = 343 and 10 × 10 × 10 = 1000.
Scale the length.Height = 30 × 710Height = 21 cm✓3
In plain EnglishHeights are lengths, so use the linear scale factor.
Problems in context
Two tins of paint are similar. The small tin is 12 cm tall and costs £3.20. The large tin is 18 cm tall. The price is proportional to the volume. Calculate the price of the large tin.(3 marks)
Linear scale factor.Scale factor = 1812 = 32✓1
In plain EnglishCompare the heights of the tins.
Volume scale factor.323 = 278✓2
In plain EnglishThe price goes up in the same way as the volume, so use the volume scale factor.
Scale the price.£3.20 × 278 = £10.80✓3
In plain English3.20 ÷ 8 = 0.40, and 0.40 × 27 = 10.80.
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Your turn
Type scale factors as fractions, like 3/2, or whole numbers. Questions marked (Calculator) need one; the rest don't. Stuck? Tap Need a hint?
Past paper questions
Ready for exam-style questions? Similar shapes come up in National 5 most years, usually for 3 marks.
National 5 similar shapesEvery National 5 similar shapes question in one PDF (2013 to 2025), with answers.maths.scot · PDF Open ↗ (opens in a new tab) Similar shapes: past paper questionsPast paper questions on similar lengths, areas and volumes.Dynamic Maths Open ↗ (opens in a new tab)