Problem 7 · Eight circles in a rectangle
Tangencies: rows of 3-2-3 equal circles and a touch of trigonometry.
Integer answer, at most 4 digitsA rectangle contains eight circles, all of the same radius. Some circles are tangent to each other and/or to the sides of the rectangle as shown in the figure. The horizontal side of the rectangle measures $30(\sqrt{3}-1)$ cm. How long is the vertical side?
Reasoned solution
Key idea: the circles form rows of 3–2–3. The width fixes the radius; the tangencies between rows give the height.
Let $r$ be the radius. The top row has $3$ circles tangent to each other and to both vertical sides, so the rectangle's width is $6r$:
The top-row centres sit at $x = r, 3r, 5r$ and the middle-row centres at $x = 2r, 4r$. Two tangent centres are $2r$ apart, with horizontal separation $r$; the vertical separation between rows is, by Pythagoras,
The total height is $r$ (edge to first centre) $+\sqrt{3}r + \sqrt{3}r + r$ (last centre to edge):