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

2. Extend the line segment past $Q$.

3. Erect the perpendicular to $\overrightarrow{PQ}$ at $Q$.

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. Draw an arc of the circle with center $P$ and radius $\overline{PQ}$.

6. Draw an arc of the circle with center $R$ and radius $\overline{QR}$ to find the point $S$ where it intersects the arc from the previous step such that $S\ne Q$.

7. Draw the square $PQRS$.

This construction is justified because $PS=PQ=QR=QS$, yielding that $PQRS$ is a rhombus. Since $\mathrm{\angle }PQR$ is a right angle, it follows that $PQRS$ is a square.

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