If the lengths of similar shapes are multiplied by a scale factor k, then the area is multiplied by k². The ratio of the areas is the square of the ratio of the lengths.
Check your understanding of the linear scale factor first, because area questions begin by finding it.
Why is area multiplied by k²?
Area measures two directions at once. A rectangle 2 cm by 4 cm has area 8 cm². Scale it by 3 and both dimensions triple: 6 cm by 12 cm, with area 72 cm².
The area did not become 24 cm² (8 × 3). It became 72 cm², which is 8 × 9, and 9 = 3². Every dimension contributes a factor of k, and there are two of them.
This works for any shape, including triangles and circles, as long as the shapes are similar.
The method
- Find the linear scale factor k from a pair of corresponding lengths, or from the area ratio.
- Square it to get the area scale factor k².
- Multiply the known area by k² if enlarging, and divide if going the other way.
- If you start with areas, write the ratio as a fraction and take the square root to get the length factor.
Worked example
Two similar shapes have areas 20 cm² and 45 cm². A side of the smaller shape is 8 cm. Find the corresponding side of the larger shape.
Step 1, area ratio. 45 ÷ 20 = 9/4.
Step 2, length ratio. The square root of 9/4 is 3/2, so k = 1.5.
Step 3, the side. 8 × 1.5 = 12 cm.
Step 4, check. With k = 1.5, k² = 2.25, and 20 × 2.25 = 45. The areas agree.
Going the other way. A shape has area 12 cm² and is enlarged by scale factor 2.5. The area scale factor is 2.5² = 6.25. So the new area is 12 × 6.25 = 75 cm².
The mistake to watch for
A common slip is to multiply the area by the linear scale factor itself.
Mistaken working: area = 12 × 2.5 = 30 cm².
This treats area as if it were a length. It ignores that area has two dimensions, so it grows by 2.5 × 2.5.
The correction is to square before you multiply: 12 × 2.5² = 12 × 6.25 = 75 cm². A sense check helps: a shape that is 2.5 times as long and 2.5 times as wide is obviously much more than 2.5 times the area.
Check yourself
Try these, then open each answer.
1. A shape has area 7 cm². It is enlarged with scale factor 3. Find the new area.
Show answer
Area scale factor = 3² = 9. New area = 7 × 9 = 63 cm².
2. Two similar shapes have areas 50 cm² and 18 cm². A side of the larger shape is 10 cm. Find the corresponding side of the smaller shape.
Show answer
Going from large to small, the area ratio is 18/50 = 9/25. The square root is 3/5, so the length factor is 0.6. The side is 10 × 0.6 = 6 cm.
3. On a floor plan with scale 1 : 100, a room has an area of 24 cm². Find the real area in m².
Show answer
The area scale factor is 100² = 10 000. Real area = 24 × 10 000 = 240 000 cm². Since 1 m² = 10 000 cm², that is 24 m².
Where this leads next
Volume follows the same pattern with one more dimension. Continue to relating volume change to a scale factor, then use the map scale and scale-factor practice tool to repeat the method with new numbers. Area and perimeter formulas are covered in area, perimeter and surface area.
If squared and cubed factors still blur together under time pressure, a teacher in online one-to-one Mathematics tuition can go through your working. They can spot where the extra power slips in or out.