Hades wrote:

If the inside circles are all of the same size and each center is a corner of the interiour square, and the interiour circles lie on the circumference of the outside circle, what is the area of the of the highlighted green area?

(1) The circumference of the outside circle is \(5\Pi\).

(2) The ratio of the outside circle's diameter to the radius of one of the interiour circles is \(2+\sqrt{2}:1\).

A.

(1) The circumference of the outside circle is \(5\Pi\).

large circle:

C = 2 PI R = 5 PI

R = 5/2

D = ac = ab+bf+cf = 5

Small & large circles:

redius of smal circles = ab = bd = de = cf.

If ab = x, bd = de = cf = x. Also be = ef = 2x

bf = sqrt(be^2 + ef^2)

bf = sqrt(4x^2 + 4x^2)

bf = sqrt(8x^2)

bf = 2x sqrt(2)

D = ab+bf+cf

5 = x + 2x sqrt(2) + x

5 = 2x + 2x sqrt(2)

5 = 2x [1+sqrt(2)]

x = 5/[2{1+sqrt(2)}]

Since we know diameters and redii of small and large circles and sides of the square, it is sufficient to find the area of the green colored space.

(2) The ratio doesnot help. Need absolute value. So insuff...