Now, in this problem, we are asked to find the area of the shaded region, this one here
and it's enclosed between the quarter of circle, this is the quarter of circle, and the square
this is the square. Okay, good. Now, the first thing we need to understand is that the square, the sign of the square
and the radius of the circle are equal. Now, on the right here, we have the whole circle and the little square here, and we
can see that the sign of the square is equal to the radius
So, the next question is, let's find the sign of the square
Now, let's call the sign of the square x, this is x and this is x
We are given the diagonal, the diagonal of the square, it can be this one here or the
other one. So, let's call it, this is the diagonal, and it is equal to 10 millimeters
So, let's use the Pythagorean theorem on this right triangle and write that the hypotenuse
which is the diagonal square, is equal to the sum of the square of the two sides
So, 10 squared is 100, and x squared plus x squared, that gives me 2x squared, divide
the two sides by 2, and that gives me x2 equal to 50
Now, let's keep it x2, and then we go to the area, now we find the area
So, the area we are looking for, this area here, it can be written as a, is equal to
the area of the whole square, so let's call it a squared, as, minus one quarter, this
is a quarter of a circle, remember, minus a quarter of the area of the circle
So, let's call it a quarter, this is a quarter, and this is the area of the whole circle
pi r squared. Now, r is the radius, and therefore, r is equal to x
Now, in the formula of the area of r squared, r squared is equal to x squared. Good
So, the area, r equal, now, the area of the square, it's equal to x squared
So, the area of the square is x multiplied by x, and that gives me x squared
So, this is x squared, and then the other side we have one quarter pi, now r squared
is equal to x squared, so this is x squared. Now, n is equal to, now, let's factor x squared, x squared, one minus one quarter of pi
We now substitute x2, there we found x2 equal to 50, so x2 equal to 50, one minus one over
four pi, and the approximation of the calculator gives 10.7, and the unit of the area here
is millimeter squared