Find the area of the shaded region in Fig. 12.22, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre.

Solution:

It is assumed that OAB is an equilateral triangle with 60° angles on each side.

The sector’s area is the common in both.

6 cm is the radius of the circle and,

12 cm is the side of the triangle.

The area of the equilateral triangle = (√3/4) (OA)²= (√3/40×12² = 36√3 cm²

The area of the circle (πR² ) = (22/7)×6² = 792/7 cm²

Area of the sector forming a 60° angle = (60°/360°) ×πr² cm²

= (1/6)×(22/7)× 6² cm² = 132/7 cm²

Area of the equilateral triangle + Area of the circle – Area of the sector = Area of the shaded region

= 36√3 cm² +792/7 cm²-132/7 cm²

= (36√3+660/7) cm²