A square is inscribed in a circle with radius ‘r’. What is the probability that a randomly selected point within the circle is not within the square ?

Sure, here are the steps to solve this problem:

Step 1: Understand the problem A square is inscribed in a circle. This means that the circle is circumscribing the square. The radius of the circle is given as 'r'. We need to find the probability that a randomly selected point within the circle is not within the square.

Step 2: Find the area of the circle The area of a circle is given by the formula πr². So, the area of the circle is πr².

Step 3: Find the area of the square The square is inscribed in the circle, so the diameter of the circle is the diagonal of the square. The length of the diagonal of a square is √2 times the side length. So, the side length of the square is the diameter of the circle divided by √2, which is 2r/√2 = r√2. The area of a square is the side length squared, so the area of the square is (r√2)² = 2r².

Step 4: Find the area of the circle that is not covered by the square The area of the circle that is not covered by the square is the area of the circle minus the area of the square. So, this area is πr² - 2r².

Step 5: Find the probability The probability that a randomly selected point within the circle is not within the square is the area of the circle that is not covered by the square divided by the total area of the circle. So, this probability is (πr² - 2r²) / πr² = 1 - 2/π.

So, the probability that a randomly selected point within the circle is not within the square is 1 - 2/π.