Q167
ArithmeticGeometry & Mensuration
What is the ratio of the perimeter of the shaded region to the circumference of the circle?
- a.$\dfrac{3}{4}$
- b.$\dfrac{4+\pi}{4\pi}$
- c.$\dfrac{2\pi}{4+\pi}$
- d.$\dfrac{4+\pi}{2\pi}$
Answer: (B) $\dfrac{4+\pi}{4\pi}$
The key treats the shaded region as a quadrant (quarter) of the circle. Its perimeter is two radii plus a quarter of the circumference: $2r+\tfrac{\pi r}{2} = \tfrac{r(4+\pi)}{2}$. Dividing by the circumference $2\pi r$ gives $\dfrac{4+\pi}{4\pi}$. Remember that the perimeter of a sector always includes the two radii, not only the arc. Figure-based question: the diagram is not part of this file.