Chapter 8: Playing with Constructions
Most of geometry has been about looking and reasoning. In this chapter we put pencil to paper and build geometry with our own hands. The only tools we use are a ruler (with a straight edge but no measuring marks, in classical tradition) and a compass (to draw circles and arcs of any radius). Yet from just these two simple tools we can construct an extraordinary range of shapes.
The Greeks loved this game. Their book Elements, written more than years ago, uses ruler-and-compass constructions to build everything from triangles to regular hexagons to elaborate star patterns. It is the original way geometry was learnt, and it is still the cleanest way to develop a feel for shape.
Constructions are precision work. A shaky hand, a loose compass, or a slightly worn pencil tip can throw off your figure by a millimetre , small but visible. So a few minutes of practising compass control is well spent before tackling the harder constructions.
What's inside
- Tools of the trade , ruler, compass, pencil, and how to use them.
- Circles, arcs, and basic shapes.
- Constructing line segments and copying lengths.
- Perpendiculars and angle bisectors.
- Triangles, squares, and regular hexagons.
Key results / Formula card
| Construction | What you produce |
|---|---|
| Copy a segment | Same length on a new line |
| Perpendicular bisector | A line cutting a segment in half at |
| Angle bisector | A ray cutting an angle into two equal halves |
| Perpendicular through a point | line through a chosen point |
| Equilateral triangle | All sides equal, all angles |
| Regular hexagon (inscribed) | Six equal sides , uses radius equal to side |
Two rules of compass-and-ruler:
- The ruler is used only to draw straight lines through two given points.
- The compass is used only to draw circles or arcs from a given point with a given radius (taken from an existing pair of points).
How to read this chapter
Do not read first and then construct , construct first. Place your tools by you and follow the steps with your hand. The geometry will sink into your fingers as well as your eyes.