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Square inscribed in a circle
Geometry construction using a compass and straightedge

How to construct a square inscribed in a given circle. The construction proceeds as follows:

  1. A diameter of the circle is drawn.
  2. 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.
  3. 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 step-by-step instructions

The above animation is available as a printable step-by-step 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

Non-Euclidean constructions