Tools and basics , ruler and compass

Drawing geometry by hand uses just two tools: a ruler and a compass. Knowing what each tool can and cannot do is half the battle.

Concept

The ruler

A ruler in geometry is a straight edge. It draws straight lines between two given points. Classical Greek geometry insisted that the ruler is unmarked , no centimetre or inch divisions. The ruler is for straightness, not for measurement. (In practice, school rulers have markings, but you should not use them for measuring in a "ruler-and-compass" construction.)

The compass

A compass has two arms , one with a sharp metal point, the other with a pencil tip. You set the distance between the two arms to a chosen length and then sweep the pencil around. This draws a circle whose centre is the metal point and whose radius is the set distance.

Two things you can do with a compass:

• Draw a complete circle of chosen radius around a chosen centre.
• Draw an arc (a piece of a circle) , just don't go all the way around.

A compass also lets you copy a distance from one place to another: you set the compass to the length of one segment, lift it (carefully) without changing the spread, and place the metal point somewhere new. The pencil then draws a circle of the same radius , anywhere on that circle is the same distance from the new centre.

Rules of the construction game

In a classical construction:

1. You start with some given points and lines.
2. You may draw a line through any two existing points (using the ruler).
3. You may draw a circle with centre at one existing point and passing through another existing point (using the compass).
4. New points are created only at intersections of existing lines or circles.

That's it. From these basic moves, an extraordinary variety of shapes can be built.

Practical tips

• Keep your pencil sharp. Blunt lines kill precision.
• Tighten the compass screw so the spread doesn't shift while you draw.
• Place the metal point firmly but don't push too hard , you'll tear the paper.
• Draw lightly. You can darken later. Erasing is harder than re-drawing.

Worked examples

Example 1. Draw a circle of radius about $3$ cm centred at a point $A$.

• Place a dot, label it $A$.
• Open the compass to about $3$ cm.
• Put the metal tip on $A$, sweep the pencil all the way around.

Example 2. Given two points $P$ and $Q$, copy the distance $PQ$ onto a new line starting at point $R$.

• Place the metal tip on $P$, the pencil tip on $Q$. The compass now holds distance $PQ$.
• Without changing the spread, lift and place the tip on $R$.
• Draw a small arc on the new line. The arc crosses the line at a point $S$ such that $RS = PQ$.

Example 3. Mark two points $A$ and $B$. Find a point $C$ that is the same distance from $A$ as $B$ is.

• Draw a circle centred at $A$ with radius $AB$.
• Any point on the circle is the same distance from $A$ as $B$ is.
• Pick any point on the circle and call it $C$.

Example 4. Why are we so careful not to "shift" the compass between operations?

• The compass holds a specific distance. If it shifts, the radius changes and the construction becomes wrong. Always lift gently and place gently.

Try it yourself

1. Draw a circle and label its centre $O$ and a point $P$ on the circle. What is the length $OP$ in terms of the radius?
2. Practice drawing five circles of different sizes. Vary the radius each time.
3. With your compass set to $4$ cm, draw a circle. Pick any two points on the circle , what is the distance between them at most? (Hint: think of the diameter.)
4. Draw a line segment $\overline{AB}$ and then a circle of radius $AB$ centred at $A$. Where does $B$ lie?
5. Draw two non-intersecting circles. Now draw two intersecting circles. Compare.
6. Mark two points $A$ and $B$. Draw a circle through $B$ centred at $A$. Then a circle through $A$ centred at $B$. They will intersect at two points , mark them.
7. Why can't you measure off a length of "exactly $\pi$ cm" with a classical ruler and compass? (Think about it , you'll see in higher classes.)
8. Reflect: what shapes do you think a ruler-and-compass cannot construct?

Activity

Compass workout. Spend $10$ minutes drawing circles, arcs, and lines without any specific goal. Try drawing concentric circles (all sharing the same centre), then circles whose centres lie on another circle. The aim is to make your hand confident with the compass before tackling real constructions.