If the radius of a circle is increased by 10%, its area will increase by—

Concept

Circle area is proportional to the square of its radius: A = πr2. If a linear measure is multiplied by a factor k, the corresponding area is multiplied by k2; the percentage change is (k2 − 1) × 100%.

Application

1. Let the original radius be r, so the original area is A = πr2.

2. A 10% increase makes the new radius r′ = 1.10r.

3. The new area is A′ = π(1.10r)2 = 1.21πr2 = 1.21A.

4. Thus A′ − A = 0.21A, so the area increases by 21%.

Cross-check

For an x% increase in radius, the area increase is 2x + x2/100 percent. With x = 10, this is 20 + 1 = 21%.

Therefore, the area increases by 21%.