Similarity: Length, Area and Volume Scale Factors
Similarity between two shapes means one is an enlargement of the other, so corresponding angles are equal and corresponding lengths are in a fixed ratio called the length scale factor, k. In this IGCSE topic, the consequence that carries the marks is that area and volume do not scale by k. Because area is a product of two lengths it scales by k squared, and because volume is a product of three lengths it scales by k cubed. That single relationship answers almost every question in the topic, including the ones that run backwards: given a ratio of volumes, the length scale factor is its cube root, and given a ratio of areas, the length scale factor is its square root.
Before you start
Make sure you're comfortable with these topics first:
Method
- Confirm the shapes are similar, either because the question says so or because the angles match, before using any scale factor.
- Find the length scale factor k by dividing a length on the second shape by the CORRESPONDING length on the first. Check which way round the question goes, since dividing the wrong way inverts every later answer.
- Apply the right power: multiply lengths by k, areas by k squared, and volumes by k cubed.
- Working backwards from an area ratio, take the square root to get k; from a volume ratio, take the cube root.
- For similar triangles inside one diagram, redraw them separately in the same orientation before writing the ratio, since that is where corresponding sides get mismatched.
- Check the answer is plausible in size: if the second shape is bigger, every length, area and volume must be bigger, and volumes grow much faster than lengths.
Worked example
Two similar cones have heights 6 cm and 9 cm. The smaller cone has surface area 80 cm squared and volume 96 cm cubed. Find the surface area and volume of the larger cone.
- Find the length scale factor from the heights: k = 9/6 = 1.5.
- Areas scale by k squared: k squared = 1.5 squared = 2.25.
- Surface area of the larger cone = 80 times 2.25 = 180 cm squared.
- Volumes scale by k cubed: k cubed = 1.5 cubed = 3.375.
- Volume of the larger cone = 96 times 3.375 = 324 cm cubed.
- Sense check: the larger cone is 1.5 times as tall, so a little over twice the surface area and nearly three and a half times the volume is exactly what the powers predict.
Practice questions
Try each question, then tap to reveal the answer.
Q1Two similar shapes have length scale factor 4. What is the area scale factor?Show answer
Answer: 16, because area scales by the square of the length scale factor.
Q2Two similar solids have volumes in the ratio 8 to 27. Find the ratio of their lengths.Show answer
Answer: Take cube roots: 2 to 3.
Q3Two similar rectangles have areas 20 cm squared and 45 cm squared. Find the length scale factor.Show answer
Answer: The area scale factor is 45/20 = 2.25, so k is the square root of 2.25, which is 1.5.
Q4A model car is built to a scale of 1 to 20. How many times greater is the volume of the real car?Show answer
Answer: 20 cubed = 8000 times greater.
Q5Why does area scale by the square of the length scale factor?Show answer
Answer: Area is calculated from two lengths multiplied together, so if each length is multiplied by k the area is multiplied by k times k.
Q6Two similar triangles have corresponding sides 5 cm and 12.5 cm. Find k.Show answer
Answer: 12.5/5 = 2.5.
Q7A cylinder's volume is multiplied by 64 under an enlargement. Find the length scale factor.Show answer
Answer: The cube root of 64, which is 4.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
Two similar jugs have heights 12 cm and 18 cm. The smaller jug holds 500 ml. Calculate how much the larger jug holds, and explain why the answer is not 750 ml.
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In triangle ABC, the line DE is parallel to BC, with D on AB and E on AC. AD = 4 cm, DB = 6 cm and DE = 5 cm. (a) Explain why triangles ADE and ABC are similar. (b) Find the length of BC.
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Free printable worksheet
Want more practice on paper? Download the similarity: length, area and volume scale factors worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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