Draw a square whose length is 40mm using a compass and ruler only. Measure and dimension the diagonal of the square.

Draw a line segment $\overline{AB}$ of length $4 \text{ cm}$.

At point $A$, construct a line perpendicular to $\overline{AB}$. To do this:

• Place the compass at $A$ and draw an arc that intersects $\overline{AB}$ at a point, say $P$.
• With the same compass radius, place the compass at $P$ and draw another arc.
• Place the compass at the intersection of the arc and $\overline{AB}$ (let's call it $Q$) and draw a third arc that intersects the second arc.
• Draw a straight line from $A$ through the intersection of the second and third arcs. This line is perpendicular to $\overline{AB}$.

On the perpendicular line from $A$, use the compass to mark a point $D$ such that $\overline{AD}$ is $4 \text{ cm}$ long.

• Place the compass at $D$ and draw an arc with a radius of $4 \text{ cm}$.
• Place the compass at $B$ and draw another arc with a radius of $4 \text{ cm}$.
• The intersection of these two arcs is point $C$.

Draw line segments $\overline{BC}$ and $\overline{CD}$ to complete the square $ABCD$.

For a square with side length $s$, the length of the diagonal $d$ can be found using the Pythagorean theorem: $d^2 = s^2 + s^2$.

To dimension it numerically: