Question 755114
The two dotted lines (OA and OB) and the arc AB form the border of a sector (a piece of pie) with AB as a curved border.
The area of that sector is {{{1/4}}} of the area of the circle, because the measure of arc AB is {{{90^o}}} and that is {{{1/4}}} of the {{{360^o}}} measure of the whole circumference.
The area of a circle is calculated as {{{pi*radius^2}}}.
In this case, the area of the circle is {{{pi*(sqrt(2))^2=pi*2=2pi}}} square units.
Then, the area of the sector with AB as a curved border is
{{{(1/4)(2pi)=pi/2}}} square units.
Triangle OAB is an isosceles right triangle.
The angle at O is a right angle.
The length of each leg is {{{OA=OB=sqrt(2)}}} units.
Taking one leg as the base of the triangle, the other leg would be the height, and the area of triangle OAB is
{{{(1/2)*base*height=(1/2)*(sqrt(2))*(sqrt(2))=(1/2)*2=1}}} square units.
The area of the shaded part is the area of sector OAB minus the area of triangle OAB.
It is {{{highlight(pi/2-1)}}} square units.