Asquare is inscribed in a circle. a point in the figure is selected at random. find the probability that the point will be in the shaded region.

Let x = side length of the square and d = diameter of the circle
d also is the hypotenuse of the square

Through the pythagorean theorem, we know that
x^2 + x^2 = d^2
2x^2 = d^2

Because d = 2r, this means
2x^2 = d^2
2x^2 = (2r)^2
2x^2 = 4r^2
x^2 = 2r^2

The area of the square is x^2, which is the same as 2r^2 as shown above

The area of the circle is A = pi*r^2

Divide the area of the square over the area of the circle to get
(x^2)/(pi*r^2) = (2r^2)/(pi*r^2) = 2/pi

The exact answer, in terms of pi, is the fraction 2/pi

If you want an approximate answer, then use a calculator to get roughly 0.6366 which is about 63.66%

That they are numbers and related to each other.