 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, in figure 12.31 a square OABC is inscribed in a quadrant OPBQ. If OA is equal to 20 centimeters, find the area of the shaded region. This is the figure 12.31. First of all let us understand that area of sector of a circle is equal to theta upon 360 multiplied by pi r square where theta is the angle of the sector in degrees and r is the radius of the circle and area of square is equal to side square. Now this is the key idea to solve the given question. Let us now start with the solution. Now in the question we are given OA is equal to 20 centimeters and we have to find area of the shaded region. We know OABC is a square and OPBQ is a quadrant. Now this angle is equal to 90 degrees since the central angle of the quadrant is 90 degrees. Moreover we also know that all the angles of the square are right angles. Now to find the area of the shaded region we will subtract area of the square from the area of the quadrant. Now to find the area of the quadrant we need to find the radius of the circle. We know radius of the quadrant or we can say radius of the circle is equal to diagonal of the square that is OB. So we can join OB as shown in the figure. We know OA is equal to AB is equal to BC is equal to OC equal to 20 centimeters. We know all the sides of the square are equal. Similarly all angles of square are right angles. So angle BAO is equal to 90 degrees. Now we will consider right triangle OAB in right triangle OB. OB square is equal to AB square plus OA square by Pythagoras theorem. Now we know AB is equal to OA is equal to 20 centimeters. So substituting corresponding values of AB and OA in this expression we get OB square is equal to square of 20 plus square of 20. Now we know square of 20 is equal to 400 so we get square of OB is equal to 400 plus 400 or we can say OB square is equal to 800. Now taking square root on both the sides we get OB is equal to 20 root 2 centimeters. Now we get radius of the circle that is R is equal to OB is equal to 20 root 2 centimeters. Now we will find out area of the quadrant. We know area of sector is equal to theta upon 360 multiplied by PyR square where theta is the angle of the sector in degrees and R is the radius of the circle. Now we know theta is equal to 90 degrees. Clearly we can see quadrant is a sector and angle of this sector is 90 degrees. And radius is equal to 20 root 2 centimeters. Now substituting corresponding values of theta and R in this formula we get area of quadrant OBBQ is equal to 90 upon 360 multiplied by Py multiplied by square of 20 root 2 centimeters square. Now substituting Py is equal to 3.14 in this expression we get area of quadrant OBBQ is equal to 90 upon 360 multiplied by 3.14 multiplied by 20 root 2 multiplied by 20 root 2 centimeters square. Now simplifying we get 1 upon 4 multiplied by 3.14 multiplied by 800 centimeters square is equal to area of quadrant OBBQ. Now this is further equal to 6 to 8 centimeters square. Now we will find out area of square OABC. We know area of square OABC is equal to 20 multiplied by 20 centimeters square. We know area of square is equal to side into side. So here side of the square is equal to 20 centimeters. So we get area of square is equal to 400 centimeters square. Now we have to find out area of shaded region. So we will subtract area of square OABC from area of quadrant OBBQ to find the area of the shaded region. Now we can write area of shaded region is equal to area of quadrant OBBQ minus area of square OABC. Now substituting corresponding values of area of quadrant and area of square OABC. In this expression we get area of shaded region is equal to 6 to 8 minus 400 centimeters square. Now we get area of shaded region is equal to 228 centimeters square. So our required answer is 228 centimeters square. This completes the session. Hope you understood the solution. Take care and keep smiling.