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 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
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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