compass and straightedge construction of regular pentagon


One can construct a regularPlanetmathPlanetmathPlanetmath (http://planetmath.org/RegularPolygon) pentagonMathworldPlanetmath 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.

    PQ
  2. 2.

    Extend the line segment past Q.

    .PQ
  3. 3.

    Erect the perpendicularMathworldPlanetmathPlanetmathPlanetmath to PQ at Q.

    ...PQ
  4. 4.

    Using the line drawn in the previous step, mark off a line segment of length 2s such that one of its endpoints is Q. Label the other endpoint as R.

    ...PQR
  5. 5.

    Connect P and R.

    ...PQR
  6. 6.

    Extend the line segment PR¯ past P.

    ...PQR
  7. 7.

    On the extension, mark off another line segment of length s such that one of its endpoints is P. Label the other endpoint as S.

    ...PQRS
  8. 8.

    Construct the midpointMathworldPlanetmathPlanetmathPlanetmath of the line segment RS¯. Label it as M. (Below, PS¯ is drawn in red, and MR¯ is drawn in green.)

    ....PQRSM

    Note that the length of the line segment MR¯ is 1+52s, which is the length of each diagonalMathworldPlanetmath of a regular pentagon with sides of length s.

  9. 9.

    Separately from the drawing from the previous steps, draw a line segment of length s.

  10. 10.

    Adjust the compass to the length of MR¯ and draw an arc from each endpoint of the line segment from the previous step so that the arcs intersect.

  11. 11.

    Adjust the compass to the length of PS¯ and draw arcs from each of the three points to determine the other two points of the regular pentagon.

  12. 12.

    Draw the regular pentagon.

The law of cosines can be used to justify this construction. Note that, in the picture below, the lengths of the line segments drawn in red are s and the lengths of the line segments drawn in green are 1+52s. The color of these line segments is based off of how the pentagon above was constructed.

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

Title compass and straightedge construction of regular pentagon
Canonical name CompassAndStraightedgeConstructionOfRegularPentagon
Date of creation 2013-03-22 17:11:12
Last modified on 2013-03-22 17:11:12
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 27
Author Wkbj79 (1863)
Entry type Algorithm
Classification msc 51-00
Classification msc 51M15
Synonym construction of regular pentagon
Related topic RegularPolygon