Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Circuscribed Square Formula
Name: Thomas
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: N/A 

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 theorem:
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

Vince Calder


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
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

Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory