A7. The diagram shows a circle divided into three equal sectors.
What is the ratio of the length of the perimeter of one of these sectors to the length of the circumference of the circle?
Solution
Let the radius of the circle be $r$. As each sector is a third of the circle, the length of the perimeter of each sector is $\frac{2 \pi r}{3}+2 r$. Therefore, the ratio of the length of the perimeter of one of the three sectors to the length of the circumference of the circle is $\left(\frac{2 \pi r}{3}+2 r\right): 2 \pi r=\left(\frac{\pi}{3}+1\right): \pi$ or $(\pi+3): 3 \pi$.