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I have the four vertices of a rectangle. I need to find it's smallest enclosing circle. For example:

enter image description here

I need to find the radius of the circle.

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    $\begingroup$ The radius of the circle is simply half the diagonal of the rectangle: $$r=\frac{1}{2}\sqrt{a^2+b^2}$$ $\endgroup$
    – Dario
    Commented Jul 2, 2014 at 17:02
  • $\begingroup$ The circle enclosing a rectangle is unique. $\endgroup$
    – user160738
    Commented Jul 2, 2014 at 17:03

3 Answers 3

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There is only one such circle for a rectangle. If your rectangle has sidelengths $a$ and $b$, then the length of the diagonal (by the Pythagorean theorem) is $\sqrt{a^2+b^2}$. Since the diagonal is a diameter, the radius is just $\dfrac{\sqrt{a^2+b^2}}{2}$.

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Take half of the distance between the endpoints of a diagonal of the rectangle.

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While pythagorean theorum works well for enclosing a square with a circle and predicting the diameter, I am concerned that it does not work for a rectangle where the hypotenuse is not perpendicular to the tangent of the circle at its end points. Is this a valid concern?

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  • $\begingroup$ Are you sure that the hypotenuse isn't perpendicular to the tangent? $\endgroup$ Commented Jun 14, 2019 at 0:08

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