Define the relationship

The surface area of a sphere is given by $S = 4\pi r^2$. Since $4\pi$ is a constant, the surface area $S$ is directly proportional to the square of the radius, $S \propto r^2$.

$S = 4\pi r^2$

Apply percentage change formula

When a quantity changes by $x\%$ and then by $x\%$, the total percentage change is calculated using the successive increase formula: $x + y + \frac{x \times y}{100}$. Here, $x = 50$ and $y = 50$.

$50 + 50 + \frac{50 \times 50}{100}$

Calculate the result

Adding the terms gives $100 + \frac{2500}{100}$, which simplifies to $100 + 25 = 125\%$.

$100 + 25 = 125\%$