In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in Fig. 12.24. Find the area of the design (shaded region).

Radius of the circle "R" = 32 cm

Draw a median AD of the triangle passing through the centre of the circle.
⇒ BD = AB/2

Since, AD is the median of the triangle

∴ AO = Radius of the circle = 2/3 AD [ By the property of equilateral triangle inscribed in a circle]

⇒ 2/3 AD = 32 cm

⇒ AD = 48 cm
In ΔADB,

By Pythagoras theorem,

AB² = AD²  + BD² 

⇒ AB² = 48²  + (AB/2)²

⇒ AB² = 2304  + AB²/4
⇒ 3/4 (AB²) = 2304

⇒ AB² = 3072

⇒ AB = 32√3 cm
Area of ΔABC = √3/4 × (32√3)² cm²
 = 768√3 cm²

Area of circle = π R² 
= 22/7 × 32 × 32
= 22528/7 cm²

Area of the design = Area of circle - Area of ΔABC

                              = (22528/7 - 768√3) cm²