You are here
Homecompass and straightedge construction of regular triangle
Primary tabs
compass and straightedge construction of regular triangle
One can construct a regular triangle 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 $Q$.
2. 3. Draw an arc of the circle with center $Q$ and radius $\overline{PQ}$ to find a point $R$ where it intersects the arc from the previous step.
4. Draw the regular triangle $\triangle PQR$.
This construction is justified by the following:

$\overline{PQ}\cong\overline{PR}$ since they are both radii of the circle from step 2;

$\overline{PQ}\cong\overline{QR}$ since they are both radii of the circle from step 3;

Thus, $\triangle PQR$ is an equilateral triangle;

In Euclidean geometry, any equilateral triangle is automatically a regular triangle. Therefore, $\triangle PQR$ is a regular triangle.
This construction is based off of the one that Euclid provides in The Elements as the first proposition of the first book. Please see this post for more details.
This construction also yields a method for constructing a $60^{{\circ}}$ angle using compass and straightedge.
Note that, with the exception of actually drawing the sides of the triangle, only the compass was used in this construction. Since regular triangles tessellate, repeated use of this construction provides a way to find infinitely many points on a line given two points on a line using just a compass.
If you are interested in seeing the rules for compass and straightedge constructions, click on the link provided.
Mathematics Subject Classification
51M15 no label found5100 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections