# What is the LCM of a and b if $(a+2 \sqrt{2} )/b$ is the ration of the area of larger circle and smaller circles?

Radius of all four smaller circles is $$R$$. If the ratio between the area of the larger circle and the sum of areas of the smallest circles is $$(a+2 \sqrt{2} )/b$$ then find the LCM of $$a$$ and $$b$$.

I don't know how to get the ration between the area of the larger circle and sum of the area of the smaller circles. Can anyone help me with a hint or formula so that I can overcome this problem?

Centers of small circles form a square with side $$2R$$. Diameter of the big circle is $$2R+2R\sqrt2$$. Radius of the big circle is therefore $$R(1+\sqrt2)$$. Check jmerry’s picture.

Ratio of big circle area and small circles area is:

$$\frac{R^2(1+\sqrt2)^2\pi}{4R^2\pi}=\frac{3+2\sqrt2}{4}=\frac{a+2\sqrt2}{b}$$.

$$b=\frac{4a+8\sqrt2}{3+2\sqrt2}\cdot\frac{3-2\sqrt2}{3-2\sqrt2}$$

$$b=12a-32-\sqrt2(8a-24)$$

$$b$$ has to be integer and therefore $$a=3, b=4.$$

Draw in the centers and the segments between them:

Now, can you find the distance $$OO_1$$ between the center of the big circle and the center of one of the small circles, in terms of $$R$$? Can you get the radius of the big circle from that?