A tessellation is a pattern created with identical shapes which fit together with no gaps or overlaps.
Tessellations in nature
Tessellations in nature have inspired tessellations in building, construction.
Tessellating triangles
Since the sum of the angles in a triangle is \(180^\circ\), three identical triangles can be placed along a straight line.
By copying and rotating this pattern, six identical triangles can be placed together at a point leaving no gaps.
This pattern can be repeated leaving no gaps.
This is a tessellating pattern.
Key point
All triangles tessellate.
Example
Prove that this triangle tessellates?
Answer
Three identical triangles will fit together on a straight line since:
\(37^\circ + 68^\circ + 75^\circ = 180^\circ \)
Six identical triangles will fit together at a point since:
\(37^\circ + 68^\circ + 75^\circ + 37^\circ + 68^\circ + 75^\circ = 360^\circ \)
Therefore, the triangle tessellates.
Tessellating regular polygons
Regular polygons will tessellate if the size of the angle is a factor of \(360^\circ\).
Equilateral triangles have angles of \(60^\circ\).
\(360^\circ \div 60^\circ = 6\)
6 equilateral triangles will fit together at a point with no gaps or overlaps.
Equilateral triangles tessellate.
Squares have angles of \(90^\circ\).
\(360^\circ \div 90^\circ = 4\)
4 squares will fit together at a point with no gaps or overlaps.
Squares tessellate.
Example
Do regular pentagons tessellate?
Regular pentagons have angles \(108^\circ\).
\(360^\circ \div 108^\circ = 3.333333…\)
3 pentagons leave a gap.
4 will overlap.
Regular pentagons do not tessellate.
Question
Do regular hexagons tessellate?
Answer
\(360^\circ \div 120^\circ = 3\)
3 regular hexagons will fit together at a point with no gaps or overlaps.
Regular hexagons tessellate.
Key point
Polygons with angles bigger than \(120^\circ\) will NOT tessellate.
More on 2D shapes
Find out more by working through a topic
- count6 of 6
- count1 of 6
