|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
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:
- Draw a line segment of length $s$ . Label its endpoints $P$ and $Q$ .
- Extend the line segment past $Q$ .
- Erect the perpendicular to $\overrightarrow{PQ}$ at $Q$ .
- 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 $R$ .
- Connect $P$ and $R$ .
- Extend the line segment $\overline{PR}$ past $P$ .
- On the extension, mark off another line segment of length $s$ such that one of its endpoints is $P$ . Label the other endpoint as $S$ .
- 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.)
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 $s$ .
- Separately from the drawing from the previous steps, draw a line segment of length $s$ .
- 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.
- 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.
- 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 $\displaystyle \frac{1+\sqrt{5}}{2}s$ . 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 link provided.
|
"compass and straightedge construction of regular pentagon" is owned by Wkbj79.
|
|
(view preamble | get metadata)
See Also: regular polygon
| Other names: |
construction of regular pentagon |
| Keywords: |
Euclidean geometry |
This object's parent.
|
|
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 7006 times total.
Classification:
| AMS MSC: | 51-00 (Geometry :: General reference works ) | | | 51M15 (Geometry :: Real and complex geometry :: Geometric constructions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
(2,0) \psdots(0,0)(2,0) \rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img1.png "\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}"){kind=small}
{.} \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}](http://images.planetmath.org:8080/cache/objects/9503/js/img2.png "\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}"){kind=small}
{.} \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}](http://images.planetmath.org:8080/cache/objects/9503/js/img3.png "\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}")
{.} \rput[b](2... ...rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img4.png "\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}")
{.} \rput[b](2... ...rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img5.png "\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}")
{.} \rput[a](2... ...rput[b](0,-0.4){$P$} \rput[a](2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img6.png "\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}")
{.} \rput[a](2... ...(2.3,0.2){$Q$} \rput[r](1.9,4){$R$} \rput[l](-0.8,-1.78885){$S$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img7.png "\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}")
{.} \rput[b](2... ...){$R$} \rput[l](-0.8,-1.78885){$S$} \rput[l](0.5528,1.1056){$M$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img8.png "\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}")
(1,-1.377) \psdots(-1,-1.377)(1,-1.377) \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img9.png "\begin{pspicture}(-2,-2)(2,-1) \psline[linecolor=red](-1,-1.377)(1,-1.377) \psdots(-1,-1.377)(1,-1.377) \end{pspicture}"){kind=small}
(-1.619,0.526)(-... ...\psdots(0,1.702)(-1.619,0.526)(-1,-1.377)(1,-1.377)(1.619,0.526) \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9503/js/img13.png "\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}")