Question

A metal wire, when bent in the form of an equilateral triangle of largest area, encloses an area of 484 `sqrt3` cm². If the same wire is bent into the form of a circle of largest area, find the area of this circle.

Answer 1

Let 'a' be the length of each side of an equilateral triangle formed.
Now, the area of an equilateral triangle formed = 484√3  cm²
⇒ `sqrt3/4`a² = 484√3  
⇒ a² = 4 x 484 
⇒ a = 2 x 22 = 44 cm

Then, perimeter of equilateral triangle = 3a = 3 x 44 = 132 cm

Now, length of wire = perimeter of equilateral triangle = circumference of circle

⇒ circumference of circle = 132 cm
⇒ 2πr =  132                 .....( r is radius of circle )
⇒ r = `[ 132 xx 7 ]/[ 2 xx 22 ]` = 21 cm

∴ Area of circle = πr² = `22/7` x 21 x 21 = 1386cm².