In a circular table-cover of radius 16cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in the figure. Find the area of the design (shaded region in the figure).

Given: Radius of circle = 16cm


Area of shaded region = Area of circle – Area of ΔABC


Firstly, we find the area of a circle


Area of circle = πr2




…(a)


Now, we will find the area of equilateral ΔABC



Construction:


Draw OD ⊥ BC


In ΔBOD and ΔCOD


OB = OC (radii)


OD = OD (common)


∠ODB = ∠ODC (90°)


∴ ΔBOD ≅ ΔCOD [by RHS congruency]


⇒ BD = DC [by CPCT]


or BC = 2BD …(i)


and,


Now, In ΔBOD, we have




⇒ BD = 8√3 cm


From (i), BC = 2BD ⇒ BC = 16√3 cm


Now, Area of equilateral ΔABC




= 192√3 cm2 …(b)


Therefore, Area of design = Area of circle – Area of ΔABC


[from (a) and (b)]


= 804.57 – 332.544


= 472.03cm2


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