Which tiles cover a floor? Polygon angles and tessellation
Fit regular triangles, squares, pentagons, hexagons and octagons round a point. Find each polygon's angle sum and interior angle, and discover which ones tessellate with no gaps.
Core concept
The interior angles of a polygon with n sides add up to (n − 2) × 180°; a regular polygon tessellates on its own only if its interior angle divides 360° exactly, as for triangles, squares and hexagons
Topic: Symmetry and tessellation
A tessellation is a pattern of shapes that covers a surface with no gaps and no overlaps, like the tiles on a floor or the honeycomb of bees. Where the corners of tiles meet at a point, their angle: The amount of turn between two lines that meet, measured in degrees (°). All key words must add up to exactly 360°. A polygon with n sides can be cut into n − 2 triangles from one corner, so its angles add up to (n − 2) × 180°. In a regular polygon all angles are equal, so each is (n − 2) × 180° ÷ n. If 360° divided by that angle is a whole number, copies fit round a point exactly and the polygon tessellates.
What you will learn
- Find the sum of the interior angles of a polygon.
- Find each interior angle of a regular polygon.
- Explain why some regular polygons tessellate and others do not.
Before you start
- Know that the angles of a triangle add up to 180°.
Materials
- Card tiles of regular polygons, or a sheet of honeycomb pattern
Equipment
- A protractor
Procedure
- Try the regular hexagon and record.
- Try the regular pentagon and record.
- Try all five tiles.
How this simulation is simplified
- All the tiles are regular polygons of one kind.