If two similar triangles have a linear scale factor of 3 from triangle A to triangle B, what is the scale factor for their areas?

If two similar triangles have a linear scale factor of 3 from triangle A to triangle B, what is the scale factor for their areas?

• A. 1/9
• B. 3
• C. 9
• D. 27

Answer: (C)

Areas scale with the square of the linear scale factor. If corresponding linear dimensions are multiplied by 3, each dimension grows by 3 and the area grows by 3 × 3, which is 9. So the scale factor for areas is 9. For example, if triangle A has area 1, triangle B would have area 9. The other options don’t fit: 1/9 would mean shrinking by 1/3, 3 would be the length factor itself, not the area factor, and 27 would be using a cube of the factor, which isn’t how area scales.

Areas scale with the square of the linear scale factor. If corresponding linear dimensions are multiplied by 3, each dimension grows by 3 and the area grows by 3 × 3, which is 9. So the scale factor for areas is 9. For example, if triangle A has area 1, triangle B would have area 9. The other options don’t fit: 1/9 would mean shrinking by 1/3, 3 would be the length factor itself, not the area factor, and 27 would be using a cube of the factor, which isn’t how area scales.