## A Geometric Puzzle

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Edited by Richard Walker, Sunday, 11 Oct 2020, 00:36

Here's an easy to understand geometric question, with a purely look-and-see solution. It comes from YouTuber Michael Penn but I have modified it slightly.

To be in the spirit of the traditional wooden Japanese temple offerings, called Sangahu, I have added decorative colours.

Question

If the larger square is 1 x 1 and so has area 1, what is the area of the smaller square?

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The diagonal of the inner square must be 1 (because that's the diameter of the circle), so to keep Pythagoras happy,each side must be the square root of 0.5 in length. Whatever that is, the area of the square must be that squared, which is obviously 0.5. Or have I got my squares mixed up?

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...alternatively, if you rotate the inner square through 45 degrees and draw the diagonals, the outer square can be seen as 4 equally-sized smaller squares, each of which is half in the inner square

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Both right, hats off to you! But the second solution is more elegant IMV.

Here are the relevant sketches.

The inner square is rotated:

And now we see that in each quadrant of the big square, the corresponding quadrant of the small square has exactly half the area.

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Thank you for a nice bit of maths, Richard.  I had fun working it all out - but much prefer the elegant solution