Three circles, each with a radius of 10, are mutually tangent to each other. The area enclosed by the three circles can be written as abã−cƒÎ, where a, b and c are positive integers, and b is not divisible by a square of a prime. What is the value of a+b+c?

Question

Three circles, each with a radius of 10, are mutually tangent to each other. The area enclosed by the three circles can be written as abã−cƒÎ, where a, b and c are positive integers, and b is not divisible by a square of a prime. What is the value of a+b+c?

Balaji · Answer

Join the centers of the 3 triangles to form an equilateral triangle with sides length 20.
The area enclosed by the 3 triangles will be the area of this triangle minus the area
of 3 sectors of the circles with central angles of 60 degrees. Thus the area of the
enclosed region will be

(1/2)*(20)*(20*sin(60)) - 3*(60/360)*pi*10^2 =

200*(sqrt(3) /2) - 50*pi = 100*sqrt(3) - 50*pi.

So a + b + c = 100 + 3 + 50 = 153.

Three circles, each with a radius of 10, are mutually tangent to each other. The area enclosed by the three circles can be written as ab�ã−cƒÎ, where a, b and c are positive integers, and b is not divisible by a square of a prime. What is the value of a+b+c?