Tessellations
Tiling the plane with no gaps or overlaps.
Key idea
Tiling the plane with no gaps or overlaps.
In depth
A tessellation tiles the plane with no gaps or overlaps. Regular tilings work only with triangles, squares, and hexagons, because the interior angle must divide 360°. Other shapes can tile in combination.
Worked example
Squares: 4 × 90° = 360°. Pentagons: 108° does not divide 360°, so regular pentagons cannot tile.
Real-world uses
- Laying tiles and bricks without gaps
- Designing wallpaper and the patterns in M. C. Escher’s art
Common mistake
Assuming any regular polygon will tile.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Tessellations?
Tiling the plane with no gaps or overlaps. It is a topic at the advanced level in Geometry.
Where is Tessellations used in real life?
Real-world examples: Laying tiles and bricks without gaps; Designing wallpaper and the patterns in M. C. Escher’s art.
What should I learn before Tessellations?
Make sure you are comfortable with Symmetry, the previous topic in Geometry.
What is a common mistake with Tessellations?
Assuming any regular polygon will tile.
How can I practise Tessellations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Tessellations?
Move on to Congruence Transformations next.
Related advanced topics in Geometry
- Heron’s Formula – Triangle area from three sides.
- Regular Polygon Area – Use apothem and perimeter.
- Arc Length of a Circle – Part of the circumference.
- Sector Area – Part of the circle’s area.
- Central and Inscribed Angles – An inscribed angle is half the central angle on the same arc.
- Chord and Tangent Properties – A tangent is perpendicular to the radius at the point of contact.