Constructions and Tilings

Overview

This chapter uses a ruler and compass to construct angles and shapes accurately, and explores tilings (tessellations) where shapes cover a surface without gaps or overlaps.

Key concepts

• Constructing line segments, angles and perpendicular bisectors.
• Bisecting an angle using a compass.
• Tilings cover a plane with repeated shapes.
• Regular tilings use triangles, squares or hexagons.

Important formulae

• Angles around a point in a tiling add to 360°
• Interior angle of a regular polygon = (n − 2) × 180° ÷ n
• A perpendicular forms a 90° angle

Solved example

1. Check whether squares tile the plane.
2. Interior angle of a square = 90°.
3. Four squares meet: 4 × 90° = 360°, so squares tile perfectly.

Important questions

1. How do you construct a 60° angle with a compass?
2. Do regular pentagons tile the plane? Explain.
3. What is a perpendicular bisector?
4. Find the interior angle of a regular hexagon.

Quick revision

Accurate constructions use only a ruler and compass. A shape tiles the plane when copies meet edge to edge with angles summing to 360° around each point.