Study case: using a scale factor to compare volumes In this GCSE Similar Shapes Volume Problem Tutor, I use one study case to show how length scale factor connects to volume scale factor. The problem is: two similar solid shapes have corresponding lengths in the ratio 2:5. The smaller solid has volume 48 cm^3. Find the volume of the larger solid. The answer is 750 cm^3. I like this example because students often remember that areas use the square of the scale factor, but volumes use the cube. Before solving, I ask the student to identify what kind of measurement is being compared. We are given lengths and asked about volume. Since volume is three-dimensional, the scale factor must be cubed. The length ratio from smaller to larger is 2:5, so the length scale factor is 5/2. Therefore the volume scale factor is (5/2)^3 = 125/8. This idea is the heart of the problem. Method 1 is the direct scale factor method. The smaller volume is 48. To find the larger volume, multiply by 125/8. So 48 times 125/8 = 6 times 125 = 750. The larger volume is 750 cm^3. I explain that the calculation is much easier if the student simplifies 48/8 first. This keeps the arithmetic clean and reduces errors. Method 2 is the ratio method. If the length ratio is 2:5, then the volume ratio is 2^3:5^3, which is 8:125. This means 8 parts correspond to 48 cm^3. One part is 6 cm^3, so 125 parts are 125 times 6 = 750 cm^3. Some students prefer this method because it keeps the ratio visible throughout the question. It is also helpful when the problem gives one shape's volume and asks for the other. Method 3 is the dimensional explanation. I ask the student to imagine a cuboid where every length is multiplied by 5/2. The length is multiplied by 5/2, the width is multiplied by 5/2, and the height is multiplied by 5/2. Volume uses all three dimensions, so the multiplier is 5/2 times 5/2 times 5/2. This helps students understand why volume scale factor is cubed rather than squared or left unchanged. After the three methods, I compare them. The direct scale factor method is efficient. The ratio method is clear and reliable. The dimensional explanation builds deeper understanding. I want students to see that similar shapes are not only about memorising a rule; they are about how measurements change when a shape is enlarged. The most common mistakes are multiplying by 5/2 instead of cubing it, using the area scale factor instead of the volume scale factor, reversing the scale factor, and forgetting cubic units. I use these mistakes as teaching clues. If the student uses the wrong power, we return to whether the measurement is length, area or volume. If they reverse the scale factor, we identify whether we are moving from smaller to larger or larger to smaller. This topic is important for GCSE and IGCSE because similar shape questions often combine ratio, area, volume and algebra. A question may give a volume ratio and ask for a length ratio, which requires cube roots. The same thinking also appears in density and enlargement problems. I teach students to label every scale factor clearly: length scale factor, area scale factor and volume scale factor. A useful extension is: if two similar solids have volumes in the ratio 27:64, find the length scale factor. Since 27 and 64 are cubes, the length ratio is 3:4. Another extension is to work backwards from a larger volume to a smaller volume. These extensions help students become flexible rather than dependent on one example. In one-to-one tutoring, I would finish by asking the student to make a small table: length scale factor, area scale factor and volume scale factor. I then give several quick questions and ask which power is needed. This builds confidence and prevents the common GCSE mistake of using the same scale factor for everything. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit GCSE and IGCSE Maths tutoring: https://www.mastermathstutoring.co.uk/gcse-igcse. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.