Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.5 AMC 10/12 Find the answer South Africa

The circle centre $O$ has radius $4$ cm. If the area of sector $OPQ$ is $\frac{1}{2}\pi$ , and if the probability that a point chosen randomly in the circle lies in the shaded sector is $\frac{1}{n}$ , then the value of $n$ is

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The area of the whole circle is $\pi \cdot 4^2 = 16\pi$ , but the area of the sector is $\frac{1}{2}\pi$ , so the probability is $\frac{\frac{1}{2}\pi}{16\pi} = \frac{1}{32}$

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