If the radius of a circle is increased by 10%, its area will increase by—
21%
The key principle here is that the area of a circle depends on the square of its radius, since area = π times radius squared. Because of this squaring, a percentage change in the radius does not translate into the same percentage change in area — the effect is amplified.
Let the original radius be r, so the original area is πr². Increasing the radius by 10% gives a new radius of 1.1r. The new area becomes π(1.1r)² = π times 1.21 times r² = 1.21πr².
The increase in area is therefore 1.21πr² minus πr² = 0.21πr². Expressed as a percentage of the original area, this is (0.21πr² divided by πr²) times 100 = 21%.
A quick shortcut confirms this: when a quantity changes by x% and the result depends on its square, the net change is approximately x + x + (x times x)/100 = 10 + 10 + 1 = 21%. The tempting wrong answers arise from linear thinking: choosing 10% simply copies the radius change and ignores the squaring; choosing 20% doubles the radius change but forgets the extra 1% coming from the product term; and 100% has no basis at all. The correct increase is 21%.
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