Three circles each of radius root 3cm ,are drawn touching each other externally. Another circle is drawn enclosing and touching all three. What is the area of the big circle ?

there are three small circle with radius root 3 cm and centers A, B and C respectively.

ABC will be an equilateral triangle, and its circum-center will be the circum-centre  for the bigger circle.

let O be the circum-center of the triangle ABC.

ABC is an equilateral triangle where AC=2 *root3 

∠AOC=360/3=120

since OA=OC

∠OAC=∠OCA=(180-120)/2=30 deg

,

 

area of the big circle is

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an equilateral triangle will be formed enclosing the enclosed region by the three circles. with side 2 root 3 cm .

so find the circumcenter of the triangle. the height of the triangle will be 3 cm. so, its circumcenter will be 2 cm

therefore, the radius the biggest circle will be 2+root3 cm

so, the area of the circle will be= pie(7+4root3) cm2

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