Question

An athletic track is designed in such a way that an athlete runs the 1${}^{st}$ and 3${}^{rd}$ quarters of the race around a circular track and the 2${}^{nd}$ and 4${}^{th}$ quarters around a square track. (As shown in the figure below) If the area of the entire square field is 256 sq units, what is the area enclosed by the running track? (Assume that the track touches the boundary of the field at the midpoints of the four sides and the width of the track is negligible).

• A. 32 + 16π sq units
• B. 64 + 16π sq units
• C. 64 + 32π sq units
• D. 32 + 32π sq units

Answer 1

The correct option is C

64 + 32π sq units

Using graphical division, we can divide the figure to look like the following:

Area of each of the 4 smaller squares = 64 sq units

Are of the shaded region in square 2 & 4 = $\frac{1}{2}$ of 64= 32 units

& Area of sqaure 1 & 3 (shaded region)= 32$\pi$

(This is because of the square circle ratio of 4:$\pi$. In this case, the ratio is 64: 16$\pi$. Hence, Area enclosed by the running track = 64 + 32$\pi$ sq units.