Question

ABCD is a square with diagonal AC = 56 cm. A circle and two quadrants of same radius are inscribed in it as shown in the figure. Find the shaded area.

Answer 1

Step 1: Find the area of the square

The area of a square can be calculated using the length of its diagonal with the formula:

Area of square = `1/2 d^2`, where d is the diagonal.

Given that the diagonal AC = 56 cm:

Area of square ABCD = `1/2 (56)^2`

= `1/2 (3136)`

= 1568 cm².

Step 2: Find the radius of the inscribed circle and quadrants

The diagonal of the square passes through the center of the large circle and the corners of the quadrants.

The length of the diagonal is equal to the sum of the radius of the quadrant at A, the diameter of the central circle and the radius of the quadrant at C.

AC = r + 2r + r = 4r.

56 = 4r.

`r = 56/4`

r = 14 cm.

So, the radius of the circle and each of the quadrants is 14 cm.

Step 3: Find the total area of the unshaded regions

The unshaded regions are one full circle and two quadrants.

Two quadrants of the same radius are equivalent to one semicircle. 

In this case, the two quadrants from corners A and C form a semicircle and there is a full circle in the center.

The total unshaded area is the sum of the area of the central circle and the area of the two quadrants.

Area of one circle = πr²

= π(14)²

= 196π cm².

Area of two quadrants = Area of a semicircle

= `1/2 πr^2`

= `1/2 π(14)^2`

= `1/2(196π)`

= 98π cm².

Total area of unshaded regions = Area of circle + Area of two quadrants

Total unshaded area = 196π + 98π = 294π cm².

Step 4: Find the area of the shaded region

The area of the shaded region is the total area of the square minus the total area of the unshaded regions.

Area of shaded region = Area of square – Total unshaded area.

Area of shaded region = 1568 – 294π cm².

Using the approximation `π ≈ 22/7` for a cleaner calculation:

`294π = 294 xx 22/7`

= 42 × 22

= 924 cm².

Area of shaded region = 1568 – 924

Area of shaded region = 644 cm².