compass and straightedge construction of square

One can construct a square with sides of a given length s using compass and straightedge as follows:

  1. 1.

    Draw a line segmentMathworldPlanetmath of length s. Label its endpointsMathworldPlanetmath P and Q.

  2. 2.

    Extend the line segment past Q.

  3. 3.

    Erect the perpendicularMathworldPlanetmathPlanetmathPlanetmath to PQ at Q.

  4. 4.

    Using the line drawn in the previous step, mark off a line segment of length s such that one of its endpoints is Q. Label the other endpoint as R.

  5. 5.

    Draw an arc of the circle with center P and radius PQ¯.

  6. 6.

    Draw an arc of the circle with center R and radius QR¯ to find the point S where it intersects the arc from the previous step such that SQ.

  7. 7.

    Draw the square PQRS.


This construction is justified because PS=PQ=QR=QS, yielding that PQRS is a rhombusMathworldPlanetmath. Since PQR is a right angleMathworldPlanetmathPlanetmath, it follows that PQRS is a square.

If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.

Title compass and straightedge construction of square
Canonical name CompassAndStraightedgeConstructionOfSquare
Date of creation 2013-03-22 17:19:13
Last modified on 2013-03-22 17:19:13
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 5
Author Wkbj79 (1863)
Entry type Algorithm
Classification msc 51M15
Classification msc 51-00