A circle is inscribed in a square with a side length of 132. If a point in the square is chosen at random, what is the probability that the point is outside the circle?
Question
A circle is inscribed in a square with a side length of 132. If a point in the square is chosen at random, what is the probability that the point is outside the circle?
Solution
Step 1: Find the area of the square. The area of a square is given by the formula side^2. So, the area of the square is 132^2 = 17424 square units.
Step 2: Find the diameter of the circle. The diameter of the circle is equal to the side length of the square, which is 132 units.
Step 3: Find the radius of the circle. The radius of a circle is half its diameter, so the radius of the circle is 132/2 = 66 units.
Step 4: Find the area of the circle. The area of a circle is given by the formula πr^2. So, the area of the circle is π(66)^2 = 13684.777 square units.
Step 5: Find the area of the square that is not covered by the circle. This is the area of the square minus the area of the circle, which is 17424 - 13684.777 = 3739.223 square units.
Step 6: Find the probability that a point chosen at random is outside the circle. This is the area outside the circle divided by the total area of the square, which is 3739.223/17424 = 0.2146 or 21.46%.
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