The scale factor k of an enlargement is image length divided by object length. Lengths are multiplied by k, and areas are multiplied by k².
This lesson is part of the section on congruency, enlargement and combined transformations. It follows recognising congruent figures.
How do I calculate k?
Use one pair of matching lengths. Put the image length on top and the object length below.
- Pick two matching sides, one on the object and one on the image.
- Divide image by object.
- Check the answer with a second pair of matching sides.
Worked example: a rectangle
A rectangle 4 cm by 3 cm is enlarged to a rectangle 10 cm by 7.5 cm.
k = 10 ÷ 4 = 2.5. Check with the other side: 7.5 ÷ 3 = 2.5, so both agree.
The object area is 4 × 3 = 12 cm². The image area is 10 × 7.5 = 75 cm².
Check against k²: 2.5² = 6.25, and 12 × 6.25 = 75. The area scale factor matches.
Worked example: a reduction
A segment of 9 cm becomes 6 cm after a reduction. k = 6 ÷ 9 = 2/3, which is less than 1.
An area of 27 cm² on the object becomes 27 × (2/3)² = 27 × 4/9 = 12 cm².
The mistake that flips the ratio
A common slip is to divide object by image, giving 4 ÷ 10 = 0.4 for the rectangle. The working looks reasonable, but the value is less than 1 and the figure clearly got bigger.
The sense check is to compare sizes first. If the image is bigger, k must be greater than 1. A second slip is to multiply area by k instead of k².
Check yourself
A triangle with a base of 6 cm is enlarged so the base becomes 15 cm. The object area is 18 cm². Find k and the image area.
Answer
k = 15 ÷ 6 = 2.5.
Image area = k² × 18 = 6.25 × 18 = 112.5 cm².
Check: the image is bigger, so k > 1 and the area grows more than the lengths do.
What to study next
Next, find where an enlargement is centred in finding the centre of enlargement. You can test coordinates in the transformation coordinates explorer.
If you want a teacher to check the direction of your ratios, see online one-to-one Mathematics tuition.