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Name: Thomas
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Question:
What is the equation that will give me the maximum square that will fit into any given circle?


Replies:
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 theorem:
L^2 + L^2 = D^2 = (2R)^2
Then:
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

Vince Calder

Hi,

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

Thomas,

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
Physics Professor
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).

Tim Mooney


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