Area and volume scale factors
When a shape is enlarged or reduced by a scale factor k, lengths change by k. Area and volume do not change by the same amount as length, so it helps to use the power rule for scaling.
1) For area
- If all lengths are multiplied by
k, then area is multiplied by k^2.
- Example idea: if the linear scale factor is 3, the area scale factor is
3^2 = 9.
2) For volume
- If all lengths are multiplied by
k, then volume is multiplied by k^3.
- Example idea: if the linear scale factor is 2, the volume scale factor is
2^3 = 8.
3) How to solve exercises
- Identify the given scale factor for lengths.
- Square it for area problems, or cube it for volume problems.
- Simplify the result completely.
4) If the scale factor is a fraction
- Treat it the same way: square or cube the fraction.
- Example:
(1/2)^2 = 1/4, (3/4)^3 = 27/64.
5) Quick check
Make sure the answer matches the type of quantity:
- area should be based on the square of the linear factor;
- volume should be based on the cube of the linear factor.
If the original shape is enlarged, the factor should be greater than 1. If it is reduced, the factor should be between 0 and 1.