The area of a circle is doubled. By what percent does its radius grow? Take √2 = 1.414.

Area grows as $r^2$, so doubling the area multiplies the radius by √2 = 1.414. The growth = 1.414 − 1 = 0.414, that is 41.4%. (Doubling the area does not double the radius — a radius twice as long would give four times the area, so 100% is far too much.)