Papy

Similar Shapes - Edexcel GCSE Higher Maths

Typically 3-6 marks per paper, usually one similar-triangle question and one similar-solids question · Spec 1MA1

Two shapes are mathematically similar when one is an enlargement of the other, so every pair of corresponding angles is equal and every pair of corresponding sides is in the same ratio. Edexcel puts this at G19 on the 1MA1 specification: apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures. The lengths part is common content, which is why a Foundation candidate can be asked for a missing side in a pair of similar triangles. The areas and volumes part is Higher only, and it is where the topic earns its reputation. Multiply every length by a scale factor kk and the area multiplies by k2k^2, while the volume multiplies by k3k^3.

Questions arrive in two families. The first is a pair of triangles, described either as one triangle sitting inside another with a line drawn parallel to the third side, or as two triangles meeting where a pair of straight lines cross. Those run to 3-4 marks and turn on pairing up the right sides before any arithmetic starts. The second is a pair of similar solids, where you are handed a surface area or a volume and asked for the matching quantity on the other solid, often through a ratio such as surface area of A : surface area of B = 9 : 16. That version is 3-5 marks and sits in the back half of the paper. Most of it lands on Papers 2 and 3, because cube rooting a ratio like 1836 : 4352 is not a Paper 1 calculation, but Edexcel does set the non-calculator version with ratios such as 8 : 27 where every root comes out whole. Nothing from G19 is printed on the Higher Tier Exam Aid, so the kk, k2k^2, k3k^3 relationship is pure recall.

Three things decide how many of these marks survive. Carrying the length scale factor into an area or a volume is the error mark schemes are built around: doubling the height of a solid and doubling its volume loses every accuracy mark, since the volume goes up eight times. Going backwards is the mirror image, and a candidate who meets an area ratio of 9 : 16 and halves it to 4.5 : 8 rather than square rooting it to 3 : 4 has thrown away the same marks. Direction is the third: write down which solid is the larger one before you multiply, because the working for 128 cm3^3 and the working for 54×276454 \times \frac{27}{64} look identical until you check the answer is on the right side of the one you started with. Where a question says 'show that triangle ABC is similar to triangle ADE', two pairs of equal angles with a named reason each is what the mark scheme wants, not a sentence about the shapes looking alike.

Worked example

P and Q are two mathematically similar solid glass paperweights. The surface area of P is 45 cm2^2 and the surface area of Q is 80 cm2^2. The volume of P is 54 cm3^3. Work out the volume of Q.
[4 marks]
  1. Both given quantities are areas, so start with the area scale factor from P to Q: 8045=169\frac{80}{45} = \frac{16}{9}.
    M1: area scale factor formed from the two surface areas, taken the right way round for P to Q
  2. The area scale factor is k2k^2, so the length scale factor is k=169=43k = \sqrt{\frac{16}{9}} = \frac{4}{3}.
    M1: square rooting to get back to a length ratio; halving 16 : 9 is the slip this mark is placed to catch
  3. Cube that to reach the volume scale factor: k3=(43)3=6427k^3 = \left(\frac{4}{3}\right)^3 = \frac{64}{27}.
    M1: cubing the length scale factor, not the area one
  4. Volume of Q =54×6427=2×64=128= 54 \times \frac{64}{27} = 2 \times 64 = 128 cm3^3.
    A1: 128 cm3^3; Q has the larger surface area, so a volume smaller than 54 would be a direction error

Practice questions

These are original questions in Edexcel 1MA1 Higher style. There are no diagrams, so each figure is set out in words: which point lies on which side, which lines are parallel, and which vertex of one triangle corresponds to which vertex of the other. Read the correspondence before you form a ratio. Several questions use ratios whose roots are exact so they work as non-calculator practice for Paper 1, and where a rounding is stated, keep the full value on your calculator until the last line.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [1 mark]
    Two mathematically similar solids have a volume scale factor of 27. What is the length scale factor?
    Answer options for question 1
  2. Question 2 Exam pace [3 marks]
    Cone P and cone Q are mathematically similar. Cone P has slant height 6 cm and total surface area 84 cm2^2. Cone Q has slant height 15 cm. Work out the total surface area of cone Q, in cm2^2.
  3. Question 3 Exam pace [2 marks]
    Which of these pairs of shapes must always be mathematically similar?
    Answer options for question 3
  4. Question 4 Stretch [4 marks]
    In triangle ABC, the point D lies on AB and the point E lies on AC. DE is parallel to BC, so triangle ADE and triangle ABC are mathematically similar, with D corresponding to B and E corresponding to C. AD = 8 cm, AB = 12 cm, AE = 10 cm and BC = 21 cm. Work out the lengths AC and DE, in cm.
  5. Question 5 Stretch [4 marks]
    Two solid spheres are made from the same material. The smaller sphere has a mass of 54 g. The larger sphere has a mass of 128 g and a radius of 8 cm. Work out the radius of the smaller sphere, in cm.
  6. Question 6 Stretch [5 marks]
    Three solids A, B and C are mathematically similar. Their heights are in the ratio 2 : 3 : 5. The surface area of B is 90 cm2^2 and the volume of C is 500 cm3^3. Work out the surface area of A, in cm2^2, and the volume of B, in cm3^3.

Common mistakes examiners see

  • Using the length scale factor on an area or a volume. Two similar cylinders have heights 3 cm and 6 cm, and the smaller holds 20 cm3^3, so the larger is answered as 40 cm3^3 instead of 20×23=16020 \times 2^3 = 160 cm3^3.

    Write the three scale factors in a column before you touch the numbers: kk, k2k^2, k3k^3. Then label the quantity in the question as a length, an area or a volume and pick the matching row. Surface area and capacity are the two words that most often get treated as lengths.

  • Dividing instead of rooting when the ratio runs the other way. An area ratio of 9 : 16 becomes a length ratio of 4.5 : 8, or a volume ratio of 27 : 64 becomes 9 : 21.3, and every later step inherits the error.

    Square root to come down from areas, cube root to come down from volumes. Check the pair you get: 3 : 4 squares back to 9 : 16, so the check takes one line. Ratios in Edexcel questions are almost always chosen so the roots are whole numbers, and a root that is not exact is usually a sign the ratio has been read as the wrong type.

  • Multiplying when the answer should be smaller. A statue 150 cm tall has a volume of 54 000 cm3^3, and the volume of the similar 50 cm model is given as 54 000 multiplied by 27 rather than divided by it.

    Before calculating, write one line saying which shape is bigger and whether your answer should be bigger or smaller than the figure you were given. If the scale factor you formed is less than 1 and your answer grew, the fraction is upside down.

  • Pairing a part with a whole in nested triangles. With D on AB, E on AC and DE parallel to BC, writing ADDB=DEBC\frac{AD}{DB} = \frac{DE}{BC} instead of ADAB=DEBC\frac{AD}{AB} = \frac{DE}{BC}.

    Redraw the small triangle separately from the large one so the sides that share a vertex are visible as whole sides. DE sits opposite A in the small triangle and BC sits opposite A in the large one, so the sides that pair with them are AD and AB, both measured from A. Add DB to AD to get AB before you form any ratio.

  • Answering a 'show that these triangles are similar' question by comparing side lengths that were never given, or by writing that the triangles are the same shape.

    Give two pairs of equal angles and a reason for each, for example angle ACB = angle ECD because vertically opposite angles are equal, and angle BAC = angle DEC because alternate angles on parallel lines are equal. Two pairs is enough, since the third pair then follows from the angle sum of a triangle, and the reasons are what carry the marks.

Frequently asked questions

Are similar shapes on the Foundation paper as well as Higher?
Partly. G19 covers both congruence and similarity, and the length side of it is common content, so a Foundation candidate can be asked to find a missing side in a pair of similar triangles. The relationships between areas and volumes in similar figures are Higher only, which is why the questions about surface area ratios and volume ratios never appear on a Foundation paper.
How do you find the volume scale factor from the area scale factor?
Square root the area scale factor to get the length scale factor, then cube it. An area scale factor of 16 gives a length scale factor of 4 and a volume scale factor of 64. Doing it in one move is what causes the standard error, because 16 cubed is 4096 and that is not the answer to anything in the question.
Is the area and volume scale factor rule given on the Edexcel formulae sheet?
No. The Higher Tier Exam Aid issued with 1MA1 papers carries the trapezium, prism and circle formulae, the quadratic formula, Pythagoras, the trigonometric ratios and rules, compound interest and the two probability rules. Nothing from G19 is on it, so if the length scale factor is kk you need to recall that areas scale by k2k^2 and volumes by k3k^3.
How many marks are similar shapes worth on Edexcel Higher?
A single missing length in a pair of similar triangles is usually 2-3 marks. A question that moves between a surface area and a volume, or works back from a ratio to a length, is typically 3-5 marks and appears in the second half of the paper. Across a full series expect roughly 3-6 marks per paper, and similarity also turns up inside other topics, most often when a frustum question needs the radius of the cone that was cut off.

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