 Hello and welcome to the session. Let us discuss the following question. Question says, from each corner of a square of side 4 cm, a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in figure 12.23. Find the area of the remaining portion of the square. This is the given figure 12.23. 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 sector in degrees and r is the radius of the circle. Also, area of square is equal to side square area of circle is equal to pi r square where r is the radius of the circle. Now we will use these formulas as our key idea to solve the given question. Let us now start with the solution. Now clearly we can see a b c d is a square whose each side is equal to 4 cm. So we can write a b is equal to b c is equal to c d is equal to d a is equal to 4 cm. We are also given that these are four quadrants of a circle of radius 1 cm. Diameter of this circle is equal to 2 cm. Now we have to find area of the shaded region. Clearly we can see if we subtract area of this circle and area of these four quadrants from area of square a b c d then we get the area of shaded region. So first of all we will find area of square a b c d. From key idea we know area of square is equal to side square each side of this square is equal to 4 cm. So area of square a b c d is equal to square of 4 cm square which is further equal to 16 cm square. So area of square a b c d is equal to 16 cm square. Now we know quadrant is nothing but a sector whose angle is 90 degrees. So we will find area of quadrant by using the formula for area of sector. We know area of sector is equal to theta upon 360 multiplied by pi r square. Now area of quadrant is equal to 90 upon 360 multiplied by pi multiplied by square of 1 cm square. We know here theta is equal to 90 degrees and radius is equal to 1 cm. Now we will substitute 22 upon 7 for pi and we get 90 upon 360 multiplied by 22 upon 7 multiplied by square of 1 cm square is equal to area of quadrant. Now simplifying further we get 90 upon 360 multiplied by 22 upon 7 multiplied by 1 cm square. We know square of 1 is equal to 1. Now 0 will get cancelled by 0. We know 9 multiplied by 1 is equal to 9 and 9 multiplied by 4 is equal to 36. So we get 1 upon 4 multiplied by 22 upon 7 cm square is equal to area of 1 quadrant. Now we know these are the 4 quadrants of a circle of radius 1 cm. So all of them have equal areas. So we can write area of 4 quadrants is equal to 4 multiplied by 1 upon 4 multiplied by 22 upon 7 cm square. Now 4 will get cancelled by 4 and we get area of 4 quadrants is equal to 22 upon 7 cm square. Now we will find out area of this circle. We know area of circle is equal to pi r square where r is the radius of the circle. Here we are given diameter of circle is equal to 2 cm. Now we know radius is equal to half of diameter. So radius of circle that is r is equal to 1 cm. Now area of given circle is equal to pi multiplied by square of 1 cm square. Now substituting 22 upon 7 for pi we get area of circle is equal to 22 upon 7 multiplied by 1 multiplied by 1 cm square. Now we get area of given circle is equal to 22 upon 7 cm square. Now we know area of shaded region is equal to area of square ABCD minus area of 4 quadrants plus area of given circle. So we can write area of shaded region is equal to area of ABCD minus area of circle plus area of 4 quadrants. Now substituting corresponding values of area of square ABCD, area of circle and area of 4 quadrants we get 16 minus 22 upon 7 plus 22 upon 7 is equal to area of shaded region. Now simplifying further we get 16 minus 44 upon 7 cm square is equal to area of shaded region. Now subtracting these two terms by taking their LCM we get 68 upon 7 cm square. So we get area of this shaded region is equal to 68 upon 7 cm square. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.