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 a√b−cπ, 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?
maths - Steve, Wednesday, February 13, 2013 at 12:42pm
Draw lines joining the centers of the circles. That is an equilateral triangle with side 10.
Area of triangle is 1/2 * 5 * 5√3 = 25/2 √3
Now, the angles of the triangle are all π/3, so, since the area of a sector of a circle is a = 1/2 r^2 θ, each sector has area 1/2 * 25 * π/3 = 25/6 π
So, the area in between all the circles is the area of the triangle, less the total area of the 3 sectors.
I think you can take it from here.
maths - correction - Steve, Wednesday, February 13, 2013 at 12:43pm
Oops. I was thinking diameter of 10, not radius. Adjust calculations accordingly.