Wikipedia:Reference desk/Archives/Mathematics/2023 August 6

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August 6

Radius of rounded corners on a square

This is not for homework; this is so I can 3D print a surrounding fitting a square with rounded corners. I believe two dimensions define the shape but I don't know how to determine the radius of those corners. (I find in practice it is difficult to accurately measure the length of the straight parts of the sides. If I could do that accurately I would know the radius of the circles.)

My two dimensions are 95 mm for the sides of the square, and 123.5 mm diagonally across the rounded corners (where the diagonal of the full square would be (95mm**2 +95mm**2)**-2 [EDITED CORRECTION: (95mm**2 +95mm**2)**1/2] or about 134 mm. Thus each rounded-off corner loses 1/2 x (134-123.5) or about 5.25 mm.

So now I am trying to find the radius of a circle that is 5.25 mm less than half the side of the square it fits exactly in. I think this means:

r - (r**2+r**2)**-2 = 5.25, [EDITED CORRECTION: r - (r**2+r**2)**1/2 = 5.25]

... but is there a simpler solution I am missing?

Please show me how to solve for r OR show me where I've gone wrong. Ideally please show me a nice general solution for the original problem, which I will re-state as: find the radius of the circles that round-off a square where the diagonal of the rounded-of square is A and the side of the whole square is B.

PS I judge that the actual answer must be roughly 11mm. Hayttom (talk) 12:41, 6 August 2023 (UTC)

Your notation is somewhat off (the exponent corresponding to the square root is 1/2, not −2), but you're doing the numbers correctly. The equation you need to solve is the other way round: ${\sqrt {2r^{2}}}-r=({\sqrt {2}}-1)r=5.25$, therefore $r=12.67$. --Wrongfilter (talk) 13:01, 6 August 2023 (UTC)

Aha! Thank you! Hayttom (talk) 14:37, 6 August 2023 (UTC)

Resolved