# What is the size of each side of the square?

The diagram shows 12 small circles of radius 1 and a large circle, inside a square.

Each side of the square is a tangent to the large circle and four of the small circles.

Each small circle touches two other circles.

What is the length of each side of the square?

Join the center of the bigger circle (radius assumed to be $$rr$$) to the mid-points of the square. It’s easy to see that $$ABCDABCD$$ is a square as well. Now, join the center of the big circle to the center of one of the smaller circles ($$PP$$). Then $$BP=r+1BP=r+1$$. Further, if we draw a vertical line through $$PP$$, it intersects $$ABAB$$ at a point distant $$r−1r-1$$ from $$BB$$. Lastly, the perpendicular distance from $$EE$$ to the bottom side of the square is equal to $$AD=rAD=r$$. Take away three radii to obtain $$EP=r−3EP=r-3$$. Using Pythagoras’ Theorem, $$(r−1)2+(r−3)2=(r+1)2r2−10r+9=0⟹r=9,1(r-1)^2 +(r-3)^2 =(r+1)^2 \\ r^2-10r+9=0 \implies r=9,1$$, but clearly $$r≠1r\ne 1$$, and so the side of the square is $$2r=182r=18$$.