Circuscribed Square Formula
What is the equation that will give me the maximum square
that will fit into any given circle?
I will try to convey the geometric construction that makes use of the Pythagorean
theorem to give the answer to your inquiry. Draw a circle of radius R. Its
diameter, D = 2R. Let 'D' be the diagonal of the square inscribed by the circle.
Note: not only is this the "maximum" square, it is the only square inscribed by
the circle. Let the side of the square be called 'L'. By the Pythagorean
L^2 + L^2 = D^2 = (2R)^2
2*L^2 = 4*R^2
Dividing both sides by '2':
L^2 = 2*R^2
Taking the square root, then:
L = (2)^1/2 * R = 1.414... * R
The area of the largest square that fits in a circle of radius R is 2R^2. This
area is about (2/Pi=) 64% of the area of the circle.
Ali Khounsary, Ph.D.
Argonne National Laboratory
The maximum square that fits into a circle is the square whose diagonal is also
the circle's diameter. The length of a square's diagonal, thanks to Pythagoras,
is the side's length multiplied by the square root of two. Set this equal to the
circle's diameter and you have the mathematical relationship you need.
Dr. Ken Mellendorf
Illinois Central College
The diagonal of the largest square that fits into a circle is equal to the
diameter 'd' of the circle, so the square has sides of length
a = d/sqrt(2).
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Update: June 2012