# Find if one rectangle can fit inside the other

Given two rectangles $R_1(l_1, b_1)$ and $R_2(l_2, b_2)$ where $l$ and $b$ are their length and breadth respectively, how to check if $R_1$ can fit inside $R_2$ or vice versa.

If $R_1$ and $R_2$ lie in the same plane and there exists an orientation of $R_1$ such that it lies completely inside $R_2$, then $R_1$ fits inside $R_2$.

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Nice question. The rectangle $R_1$ rotated by $\theta$ radians fits in an axis-aligned box of length $l_1\lvert\cos\theta\rvert+b_1\lvert\sin\theta\rvert$ and width $l_1\lvert\sin\theta\rvert+b_1\lvert\cos\theta\rvert$. So you want to determine whether there is any $\theta$ such that these are less than $l_2$ and $b_2$ respectively. Of course, one can restrict $\theta$ to $[0,\pi/2]$ and drop the absolute value signs. This leaves you with inequality constraints on a couple of trigonometric functions, and I don't feel like working it out after that... – Rahul Sep 3 '12 at 12:05
This reminds me of a story where a train allowed only items of a max length/breadth/depth, so to get his over-long umbrella onto the train the passenger put it into a box. – binn Sep 3 '12 at 12:55