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Extended Only Geometry

Similarity for IGCSE Maths

Identifying and using similar shapes, including area and volume ratios. This subtopic is part of Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding similarit

What You Need to Know

Identifying and using similar shapes, including area and volume ratios. This subtopic is part of Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding similarity is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Similarity

Two shapes are similar if one is an enlargement of the other — all corresponding angles are equal and all corresponding sides are in the same ratio. The ratio of corresponding sides is the scale factor k. For similar shapes: if linear scale factor = k, then area scale factor = k² and volume scale factor = k³. IGCSE Extended tests proving similarity, finding missing lengths, and applying the area/volume scale factor relationships.

Step-by-Step Method

  1. 1

    Prove similarity

    Show AA (two pairs of equal angles), SSS (all three sides in proportion), or SAS (two sides in proportion and included angle equal).

  2. 2

    Find the scale factor k

    k = corresponding side of larger shape / corresponding side of smaller shape. Use a pair of known corresponding sides.

  3. 3

    Find unknown lengths

    Unknown side = known corresponding side × k. Always multiply the smaller by k to get the larger.

  4. 4

    Apply area and volume ratios

    Area ratio = k². Volume ratio = k³. If areas are given, find k = √(area ratio); if volumes, k = ∛(volume ratio).

  5. 5

    Write a clear conclusion

    State the triangles are similar and the matching vertices. E.g. 'Triangle ABC is similar to triangle DEF (AA)'.

Worked Example

Question

Triangle ABC is similar to triangle DEF. AB = 6 cm, DE = 9 cm. The area of triangle ABC = 24 cm². Find the area of triangle DEF.

Solution

Scale factor k = DE/AB = 9/6 = 3/2 Area scale factor = k² = (3/2)² = 9/4 Area of DEF = 24 × 9/4 = 54 cm² Answer: Area of DEF = 54 cm²

Exam Tips for Similarity

  • Always find k from side lengths — then use k² for areas and k³ for volumes.
  • If given area ratio, take the square root to find k; if given volume ratio, take the cube root.
  • Match vertices in the correct order when stating similarity: Triangle ABC ~ Triangle PQR means A↔P, B↔Q, C↔R.
  • Parallel lines across a triangle always create similar sub-triangles — this is the most common setup in IGCSE.

Practice Questions

Q1: Two similar cones have volumes 125 cm³ and 1000 cm³. Find the ratio of their curved surface areas.

Show hint

Volume ratio = 1000/125 = 8 = k³ → k = 2. Area ratio = k² = 4.

Q2: Triangles PQR and XYZ are similar. PQ = 4 cm, QR = 6 cm, XY = 10 cm. Find YZ.

Show hint

k = 10/4 = 5/2. YZ = 6 × 5/2 = 15 cm.

Q3: In the diagram, DE is parallel to BC. AD = 3, DB = 5, DE = 4.5 cm. Find BC.

Show hint

Triangles ADE and ABC are similar (AA). k = AB/AD = 8/3. BC = 4.5 × 8/3 = 12 cm.

Frequently Asked Questions

What is similarity in IGCSE Maths?

Identifying and using similar shapes, including area and volume ratios.

Is similarity in the Core or Extended syllabus?

Similarity is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise similarity effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on similarity rather than trying to cover everything at once.

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