The side of a square is 5 cm. How many times will the new area become, if the…

Concept: For any square, Area = (side length)2. When every side of a square is scaled by a factor k, the resulting area is scaled by k2 -- not by k, which describes linear (length) scaling, and not by k3, which describes volume scaling for a 3-dimensional solid.

1. The original side of the square is 5 cm, so the original area = 52 = 25 cm2.

2. Doubling the side gives the new side length = 2 x 5 cm = 10 cm.

3. The new area = 102 = 100 cm2.

4. The ratio of the new area to the original area = 100 cm2 / 25 cm2 = 4.

Cross-check: Using the general scaling relationship directly, a linear scale factor of k = 2 gives an area scale factor of k2 = 22 = 4, which matches the ratio found by direct calculation. So the new area becomes 4 times the original area.