Similarity — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers similar shapes: shapes with the same form but different sizes. By the end of this guide you should be able to decide whether two shapes are similar, find the length scale factor, and use it to calculate a missing length.
You should also be able to scale areas and volumes correctly, work with similar triangles inside a larger figure, and set out a proof that two triangles are similar.
The organising idea is that the scale factor changes depending on what you are scaling. Lengths scale by k, areas by k² and volumes by k³ — and the reason is straightforward: an area is two lengths multiplied, so both of them grow by k, and a volume is three. Double every length of a shape and its area becomes four times larger while its volume becomes eight times larger. Nearly every mark lost in this topic comes from applying the length scale factor to an area or a volume, and holding on to "one k per dimension" prevents it.
Key terms and definitions
Similar — same shape, with equal corresponding angles and corresponding sides in the same ratio.
Congruent — identical in both shape and size, which is the special case where the scale factor is 1.
Corresponding — occupying matching positions in the two shapes.
Length scale factor (k) — the number multiplying each length of the smaller shape to give the larger.
Area scale factor — k², the factor by which areas increase.
Volume scale factor — k³, the factor by which volumes increase.
Core concepts
What makes two shapes similar
Two conditions must hold: corresponding angles are equal, and corresponding sides are in the same ratio.
For triangles, either one on its own is enough, because the angles fix the shape. Two triangles with the same three angles are automatically similar, and so are two whose three pairs of sides share a common ratio.
For other shapes both conditions matter. A square and a non-square rectangle have all angles equal at 90°, yet are plainly not similar, because their sides are not in a common ratio.
Similar shapes may be rotated or reflected relative to each other, which makes matching up the corresponding sides the first real task in most questions.
Finding the length scale factor
Divide a length on one shape by the corresponding length on the other.
If a triangle with a base of 4 cm is enlarged so the matching base is 10 cm, the scale factor is 10 ÷ 4 = 2.5.
Then multiply any other length of the small shape by 2.5 to get its partner, or divide a length of the large shape by 2.5 to go the other way.
Getting the correspondence right is essential, and it is easiest by matching angles rather than by position on the page. The side opposite the largest angle in one triangle corresponds to the side opposite the largest angle in the other.
A scale factor greater than 1 enlarges; one between 0 and 1 reduces. Both are valid, and choosing which way round to divide depends on which shape you are going to.
Scaling areas
Areas scale by k².
An area is made from two lengths multiplied together, so when each length grows by a factor of k, the area grows by k × k.
Double the lengths of a shape and its area becomes four times bigger, not twice. Triple them and the area becomes nine times bigger.
To go from an area scale factor back to a length scale factor, take the square root. If one shape's area is 25 times another's, its lengths are 5 times as long.
Scaling volumes
Volumes scale by k³, since a volume is three lengths multiplied.
Double the lengths of a solid and its volume becomes eight times bigger.
To recover the length scale factor from a volume ratio, take the cube root. If one solid's volume is 27 times another's, its lengths are 3 times as long.
Surface area, being an area, still scales by k² even on a three-dimensional solid — so a solid whose lengths double has four times the surface area and eight times the volume.
Moving between the three
Many questions give information about one measure and ask about another, and the route always goes through k.
Given an area ratio and asked for a volume ratio: square-root the area ratio to find k, then cube it.
So if the areas are in the ratio 9 : 16, then k = 4 ÷ 3, and the volumes are in the ratio 27 : 64.
Writing k down explicitly as a separate step is what makes these questions manageable.
Similar or congruent?
The two words describe different relationships, and questions test the distinction directly.
Congruent shapes are identical in shape and size, so every pair of corresponding sides is equal and the scale factor is exactly 1. Congruence is proved by one of the standard criteria: SSS, SAS, ASA or RHS.
Similar shapes share the form but not necessarily the size, so the scale factor can be any positive number.
Every pair of congruent shapes is therefore similar, but the reverse does not hold. If a question asks you to prove congruence, matching angles alone is never enough — at least one pair of equal sides is always required, which is why every congruence criterion contains an S.
Scale factors between 0 and 1
A scale factor need not enlarge. Going from the larger shape to the smaller gives a value between 0 and 1, and the same rules apply.
If k = 1/2, then areas scale by 1/4 and volumes by 1/8, so the smaller solid has an eighth of the volume.
Questions sometimes phrase this as a reduction or as a fraction, and the arithmetic is identical — only the direction changes. Deciding at the outset which shape you are scaling to settles whether k should be greater or less than 1, and prevents the commonest error of dividing when you should multiply.
Similar triangles inside a figure
A very common arrangement has one triangle sitting inside another, sharing a vertex, with a line parallel to one side cutting across.
Because the cutting line is parallel, corresponding angles are equal, so the two triangles are similar.
The difficulty is that the sides often overlap. A length running from the shared vertex may be given as the part rather than the whole, so check whether a stated length reaches all the way along the side of the larger triangle or stops partway.
Redrawing the two triangles separately, the right way up, removes almost all of that confusion. It is the same technique as extracting a right-angled triangle in three-dimensional trigonometry.
Proving two triangles are similar
A proof needs a stated reason for each pair of equal angles, and two pairs is enough — the third follows automatically, since the angles of a triangle total 180°.
The reasons are the ordinary angle facts: angles in the same segment, vertically opposite angles, corresponding or alternate angles with parallel lines, or a shared angle common to both triangles.
Set it out as a list: name the first pair of equal angles with the reason, name the second pair with its reason, then conclude that the triangles are similar.
A common omission is the shared angle. Where two triangles overlap at a vertex, that angle belongs to both and is worth stating.
Worked examples
Example 1: A missing length
Two similar triangles have corresponding sides of 6 cm and 15 cm. Another side of the smaller triangle is 8 cm. Find the matching side of the larger triangle.
The length scale factor is 15 ÷ 6 = 2.5.
Multiply the known small side by it: 8 × 2.5 = 20 cm.
Sense check: the larger triangle's side should be bigger than 8 cm, and it is. Dividing instead would have given 3.2 cm, which is the wrong direction.
Example 2: Area and volume together
Two similar cones have volumes of 54 cm³ and 128 cm³. The smaller has a curved surface area of 27 cm². Find the curved surface area of the larger.
Work through k. The volume ratio is 128 ÷ 54, and the length scale factor is its cube root.
Since 128 = 4³ × 2 and 54 = 3³ × 2, the ratio 128 : 54 simplifies to 64 : 27, whose cube roots are 4 and 3. So k = 4/3.
Surface area is an area, so it scales by k² = 16/9.
The larger area is 27 × 16/9 = 48 cm².
Applying k directly to the area would have given 36 cm², which is the standard error this topic is built to test.
Example 3: Triangles inside a figure
In a triangle, a line parallel to the base cuts the other two sides. The small upper triangle has a side of 5 cm where the whole large triangle has 12 cm along the same line. The base of the small triangle is 4 cm. Find the base of the large triangle.
The parallel line makes corresponding angles equal, so the two triangles are similar.
The scale factor is 12 ÷ 5 = 2.4.
The large base is 4 × 2.4 = 9.6 cm.
The check to make here is whether the 12 cm is the whole side or only the part below the cut. Read the figure carefully: if 12 cm were the lower portion only, the whole side would be 17 cm and the scale factor 3.4 instead.
Common mistakes and how to avoid them
Using k for areas or volumes. Areas scale by k², volumes by k³. One k per dimension.
Forgetting to root when working backwards. An area ratio needs a square root and a volume ratio a cube root to recover k.
Matching the wrong sides. Corresponding sides face corresponding angles, not matching positions on the page.
Dividing the wrong way. Check the answer's size: going to the larger shape must increase the length.
Treating surface area as a volume. It is an area, so it scales by k² even on a solid.
Misreading a part as a whole in nested triangles. Redraw the two triangles separately.
Giving a similarity proof without reasons. Each pair of equal angles needs a named reason.
Exam technique for "Similarity"
Write the length scale factor down as its own line before calculating anything else. Every other part of the question flows from it, and it usually carries a method mark.
State which scale factor you are using at each step — k, k² or k³ — so the examiner can see the reasoning.
Redraw nested triangles separately, in matching orientations, before reading lengths off.
Sense-check every answer by direction: lengths on the larger shape must be larger.
For a proof, list two pairs of equal angles with a named reason for each, then state the conclusion.
Watch for questions that mix measures — an area given and a volume wanted — and route the working through k explicitly.
Quick revision summary
Similar shapes have equal corresponding angles and corresponding sides in the same ratio. For triangles, equal angles alone are enough.
The length scale factor k comes from dividing corresponding lengths.
Lengths scale by k, areas by k², volumes by k³ — one k for each dimension. Doubling the lengths gives 4 times the area and 8 times the volume.
Working backwards, square-root an area ratio and cube-root a volume ratio to recover k.
Surface area is an area, so it scales by k² even on a solid.
For nested triangles created by a parallel line, the shapes are similar by corresponding angles — redraw them separately and check whether a given length is a part or a whole.
To prove similarity, give two pairs of equal angles, each with a named reason, then state the conclusion.