To manufacture nine circular cardboard discs for Carnival, square sheets of $10 \mathrm{~cm}$ on a side are used, as indicated in the figure. What is the area $\left(\mathrm{in} \mathrm{cm}^{2}\right)$ of the unused paper?

(a) 25

(b) 22.5

(c) 21.5

(d) 21

(e) 22
The correct option is (c).

Remember that the area of a circle with radius $r$ is $\pi r^{2}$. If $r$ is the radius of the circles in the figure, then the unused area is the area of the square, which is given by $10 \times 10=100 \mathrm{~cm}^{2}$, minus the sum of the areas of the nine circles, which is $9 \times \pi r^{2} \mathrm{~cm}^{2}$. It turns out that the radius of each circle is $r=5 / 3 \mathrm{~cm}$, since the diameters of three of these circles sum to one side of $10 \mathrm{~cm}$ of the cardboard sheet. Thus, the unused area is given by

$$
100-9 \times \pi \times\left(\frac{5}{3}\right)^{2}=100-25 \pi
$$

Using the approximation $\pi \approx 3.14$, it follows that the unused area measures

$$
100-25 \pi \approx 100-25 \times 3.14=21.5 \mathrm{~cm}^{2}
$$