In the diagram, O is the center of a circle with radii OP=OQ=15. What is the perimeter of the shaded region?
Answer: \(22.5\pi + 30\), or rounded as \(\approx 100.686\)
Pretend that the shaded region is a full circle with radius of 15. The circumference of that circle would be \(2\pi r\), or since r = 15, \(30\pi\).
Because a circle has 360 degrees, and there is 90 degrees being taken out of it, one quarter of the circle has been 'unshaded'. So you have to take away 1/4 of the \(30\pi\), which just leaves \(\frac 3 4 \cdot 30\pi\) which equals \(22.5\pi\) left. There is still one thing left. When one quarter of the circle was taken away, OQ and OP were left exposed as perimeter, which means you have to add 15 x 2 = 30 to the answer. This leads to the final answer being \(22.5\pi + 30\), or rounded as \(\approx 100.686\).