For similar solids, take the square root of the surface area scale factor to find the length scale factor. Then cube the length scale factor to find the volume scale factor. Keep the same order of objects at every step.
The 2027 SEC G3 Mathematics K310 syllabus covers area and volume ratios in similarity. This worked example starts with a surface area ratio and shows how to use it to find both a larger volume and the difference between the two volumes.
- Confirm that the solids are similar.
- Write both quantities in the same order and use matching units.
- Take the square root of the area factor, then cube the result.
- Apply the volume factor before finding any difference.
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Why the area factor comes first
When the length factor is k, every corresponding length is multiplied by k. A flat face has two dimensions, so its area is multiplied by k². All corresponding faces of similar solids scale by that same factor, so their total surface areas do too. Volume has three dimensions and scales by k³.
Choose a direction before you calculate. If the larger solid has 2.25 times the area of the smaller one, take the square root of 2.25 to find the factor for going from smaller to larger. If you reverse the comparison, use the reciprocal throughout.
Check the premise. Equal shape with proportional corresponding lengths supports this method. Surface areas alone do not determine the volume ratio of solids with unrelated shapes.
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Worked example | Surface area ratio 16 to 25
Two similar solid models, A and B, have total surface areas in the ratio 16 : 25. Model A has a volume of 256 cm³. Find the volume of B and how much more volume B has than A.
| Quantity | Operation | A : B |
|---|---|---|
| Surface area | Given | 16 : 25 |
| Length | Square root of each term | 4 : 5 |
| Volume | Cube of each length term | 64 : 125 |
The 64 parts of volume for model A equal 256 cm³, so each part is 4 cm³. Model B has 125 parts, giving 500 cm³. The difference is 500 − 256 = 244 cm³.
You can check the answer with the multiplier 125/64. This is bigger than 1, as it should be when moving from A to B. The units of 256 and 500 are both cubic centimetres. The ratio itself has no unit.
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Practise in the opposite direction
Two similar display models have volumes in the ratio 27 : 64. The larger one has a surface area of 192 cm². Find the smaller area. Cube roots give a length ratio of 3 : 4. Squaring gives an area ratio of 9 : 16.
The larger area corresponds to 16 parts. One part is 192 ÷ 16 = 12 cm². The smaller area is 9 × 12 = 108 cm². Reading the order aloud before dividing helps you match the known number to the right number of parts.
Try a direct check with a cube of side 3 cm and a cube of side 4 cm. Their areas are 54 cm² and 96 cm², which simplify to 9 : 16. Their volumes are 27 cm³ and 64 cm³.
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Frequently asked questions
Can I take the cube of the surface area ratio?
Start by recovering the length ratio. Cubing the area ratio would multiply three area factors, which scales with the sixth power of the length factor.
What if the area ratio is 2 to 3?
Use √2 : √3 as the length ratio. You can leave the roots in exact form until the final step.
What can a parent check quickly?
Ask the student to point to the same model on all three ratio lines and explain why the volume answer has cubic units.
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