Question

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

Answer 1

(i)

Central angle = 40°

Radius (r) of the circle = 5 cm

Area of the sector =

=

Area of the shaded part = Area of the sector

(ii)

Central angle = 60°

Radius (r) of the circle = 6 cm

Area of the sector =

=

_{= 6π cm}²

Area of the circle = π r²

= π (6)² cm²

= 36π cm²

Area of the shaded part = Area of the circle − Area of the sector

= (36π − 6π) cm²

= 30π cm²

(iii)

Central angle = 60°

Radius (r) of the circle = 7 cm

Area of the sector =

=

In ΔAOB, OA = OB = 7 cm

⇒ ∠OBA = ∠OAB (Angles opposite to equal sides are equal in measure.)

Using angle sum property in ΔAOB:

∠OBA + ∠OAB + ∠AOB = 180°

⇒ 2∠OBA + 60° = 180°

⇒ 2∠OBA = 120°

⇒ ∠OBA = 60°

∴ ΔOAB is an equilateral triangle.

Area of the equilateral triangle =

Area of the shaded part = Area of the sector − Area of the equilateral triangle

= (25.65 − 21.19) cm²

= 4.46 cm²

(iv)

Radius of the bigger circle = 6 cm

Radius of the smaller circle = 5 cm

Central angle made by the arcs = 120°

Area of the bigger sector =

=

_{= 12π cm}²

Area of the smaller sector =

Area of the shaded part = Area of the bigger sector − Area of the smaller sector

= (12π − 8.33π) cm²

= 3.67π cm²ⁿ

= (3.67 × 3.14) cm²

= 11.52 cm²