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[parent] compass and straightedge construction of regular pentagon (Algorithm)

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

  1. Draw a line segment of length $ s$. Label its endpoints $ P$ and $ Q$.

    \begin{pspicture}(-1,-1)(5,1) \psline[linecolor=blue](0,0)(2,0) \psdots(0,0)(2,0) \rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \end{pspicture}
  2. Extend the line segment past $ Q$.

    \begin{pspicture}(-1,-1)(5,1) \rput[r](5,0){.} \psline(0,0)(2,0) \psline[linecol... ...) \psdots(0,0)(2,0) \rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \end{pspicture}
  3. Erect the perpendicular to $ \overrightarrow{PQ}$ at $ Q$.

    \begin{pspicture}(-2,-3)(5,5) \psline{o->}(0,0)(5,0) \rput[r](5,0){.} \rput[b](2... ...dots(0,0)(2,0)(4,0) \rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \end{pspicture}
  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$.

    \begin{pspicture}(-2,-3)(5,5) \psline{o->}(0,0)(5,0) \rput[r](5,0){.} \rput[b](2... ...rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \end{pspicture}
  5. Connect $ P$ and $ R$.

    \begin{pspicture}(-2,-3)(5,5) \psline{o->}(0,0)(5,0) \rput[r](5,0){.} \rput[b](2... ...rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \end{pspicture}
  6. Extend the line segment $ \overline{PR}$ past $ P$.

    \begin{pspicture}(-2,-3)(5,5) \psline{o->}(0,0)(5,0) \rput[r](5,0){.} \rput[a](2... ...rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \end{pspicture}
  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$.

    \begin{pspicture}(-2,-3)(5,5) \psline{o->}(0,0)(5,0) \rput[r](5,0){.} \rput[a](2... ...(2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \rput[l](-0.8,-1.78885){$S$} \end{pspicture}
  8. Construct the midpoint of the line segment $ \overline{RS}$. Label it as $ M$. (Below, $ \overline{PS}$ is drawn in red, and $ \overline{MR}$ is drawn in green.)

    \begin{pspicture}(-2,-3)(5,5) \psline{o->}(0,0)(5,0) \rput[r](5,0){.} \rput[b](2... ...){$R$} \rput[l](-0.8,-1.78885){$S$} \rput[l](0.5528,1.1056){$M$} \end{pspicture}

    Note that the length of the line segment $ \overline{MR}$ is $ \displaystyle \frac{1+\sqrt{5}}{2}s$, which is the length of each diagonal of a regular pentagon with sides of length $ s$.

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

    \begin{pspicture}(-2,-2)(2,-1) \psline[linecolor=red](-1,-1.377)(1,-1.377) \psdots(-1,-1.377)(1,-1.377) \end{pspicture}
  10. Adjust the compass to the length of $ \overline{MR}$ and draw an arc from each endpoint of the line segment from the previous step so that the arcs intersect.

    \begin{pspicture}(-2,-2)(2,2) \psline(-1,-1.377)(1,-1.377) \psarc[linecolor=gree... ...(1,-1.377){3.236}{90}{125} \psdots(-1,-1.377)(1,-1.377)(0,1.702) \end{pspicture}
  11. Adjust the compass to the length of $ \overline{PS}$ and draw arcs from each of the three points to determine the other two points of the regular pentagon.

    \begin{pspicture}(-2,-2)(2,2) \psline(-1,-1.377)(1,-1.377) \psarc(-1,-1.377){3.2... ...\psdots(0,1.702)(-1.619,0.526)(-1,-1.377)(1,-1.377)(1.619,0.526) \end{pspicture}
  12. Draw the regular pentagon.

    \begin{pspicture}(-2,-2)(2,2) \psarc(-1,-1.377){3.236}{55}{90} \psarc(1,-1.377){... ...\psdots(0,1.702)(-1.619,0.526)(-1,-1.377)(1,-1.377)(1.619,0.526) \end{pspicture}

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 $ \displaystyle \frac{1+\sqrt{5}}{2}s$. The color of these line segments is based off of how the pentagon above was constructed.


\begin{pspicture}(-2,-2)(2,2) \pspolygon[linecolor=red](0,1.702)(-1.619,0.526)(-... ...\psdots(0,1.702)(-1.619,0.526)(-1,-1.377)(1,-1.377)(1.619,0.526) \end{pspicture}

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



"compass and straightedge construction of regular pentagon" is owned by Wkbj79.
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See Also: regular polygon

Other names:  construction of regular pentagon
Keywords:  Euclidean geometry

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Cross-references: compass and straightedge constructions, color, law of cosines, points, intersect, arc, diagonal, midpoint, line, erect the perpendicular, endpoints, line segment, straightedge, compass, length, sides, pentagon
There are 2 references to this entry.

This is version 24 of compass and straightedge construction of regular pentagon, born on 2007-06-03, modified 2007-06-25.
Object id is 9503, canonical name is CompassAndStraightedgeConstructionOfRegularPentagon.
Accessed 3371 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M15 (Geometry :: Real and complex geometry :: Geometric constructions)
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