Q. 63.7( 262 Votes )
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).
Answer :
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
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