Question

In each of the figures below, find the area of the shaded part

Answer 1

(i)

Construction: Join the diagonal AC of the given rectangle.

The diagonal AC will pass through the centre of the circle. So, AC is the diametre of the circle.

It can be observed that the diagonals of the regular hexagon divide it into six equilateral triangles.

In ΔADC, ∠ADC = 90°

Using Pythagoras theorem:

AC² = AD² + DC²

= (3² + 4²) cm²

= (9 + 16) cm²

= 25 cm²

Radius (r) of the circle =

Area of the circle = π × square of radius

= (3.14 × 2.5 × 2.5) cm²

= 19.625 cm²

Area of the rectangle, ABCD = Length × Breadth

= 4 × 3 cm²

= 12 cm²

Area of the shaded region = Area of the circle − Area of the rectangle, ABCD

= 19.625 cm² − 12 cm²

= 7.625 cm²

(ii)

Construction: Join the diagonals of the regular hexagon.

The diagonals of the regular hexagon pass through the centre of the circle.

It can be observed that the diagonals of the regular hexagon divide it into six equilateral triangles.

∴ ΔOAB is an equilateral triangle with OA = OB = AB = 2 cm

∴ Radius (r) of the circle = 2 cm

Area of the circle = π × square of radius

= (3.14 × 2 × 2) cm²

= 12.56 cm²

Area of the hexagon, ABCDEF =

Area of the shaded region = Area of the circle − Area of the hexagon, ABCDEF

= (12.56 − 10.38) cm²

= 2.18 cm²

(iii)

Radius of the larger circle = 2 cm

Radius of the smaller circle = 1 cm

Area of the larger circle = π × square of radius

= (3.14 × 2 × 2) cm²

= 12.56 cm²

Area of the smaller circle = π × square of radius

= (3.14 × 1 × 1) cm²

= 3.14 cm²

Area of the shaded region = Area of the larger circle − Area of the smaller circle

= (12.56 − 3.14) cm²

= 9.42 cm²

(iv)

Side of the square = 4 cm

Radius of the circle whose quadrants are shown in the figure =

Area of the square = side × side

= 4 × 4 cm²

= 16 cm²

Area of one quadrant = π × square of radius

=

_{= 3.14 cm}²

Area of the four quadrants = 4 × 3.14 cm² = 12.56 cm²

Area of the shaded region = Area of the square − Area of the four quadrants

= (16 − 12.56) cm²

= 3.44 cm²