
Square inscribed in a circle
How to construct a square inscribed in a given circle. The construction proceeds as follows:
 A diameter of the circle is drawn.
 A perpendicular bisector of the diameter is drawn using the method described in
Perpendicular bisector of a segment. This is also a diameter of the circle.
 The resulting four points on the circle are the vertices of the inscribed square.
No center point?
If the circle's center point is not given, it can be constructed using the method in
Constructing the center of a circle.
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Proof

Argument 
Reason 
1 
AC is a diameter of the circle O 
A diameter is a line through the circle center.
See Diameter definition.

2 
BD is a diameter of the circle O 
It was drawn using the method in
Perpendicular bisector of a line.
See that page for proof. The center of a circle bisects the diameter, so BD passes through the center. 
3 
AC, BD are perpendicular 
BD was drawn using the method in
Perpendicular bisector of a line.
See that page for proof. 
4 
AC, BD bisect each other 
Both are diameters of the circle O. (1), (2) and the center of a circle bisects its diameter.
See Diameter definition 
5 
ABCD is a square 
Diagonals of a square bisect each other at 90°. (3), (4) 
6 
ABCD is an inscribed square 
All vertices lie on the given circle O 
 Q.E.D
Try it yourself
Click here for a printable worksheet containing some problems to try.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
(C) 2011 Copyright Math Open Reference. All rights reserved

