Quadrilaterals, tiling and the eight directions · Usually met around P4
Cutting up and putting together
From one corner: triangles = sides − 2.
How to do it
Method
From one corner: triangles = sides − 2
Hexagon: 6 − 2 = 4 triangles
Another way
From the centre
A hexagon makes 6 triangles
In full
Cut a polygon along straight lines and you get smaller polygons; count the pieces one by one and see how many sides each has. Drawing every diagonal from one corner of a regular polygon gives 2 fewer triangles than it has sides: a hexagon gives 6 − 2 = 4 triangles; joining the centre to every corner gives as many triangles as sides. The other way round, small shapes put together make bigger ones: two equal equilateral triangles make a rhombus, and a square of side 3 needs 3 × 3 = 9 squares of side 1. Or draw each piece out and count, as a check.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Cutting the shape along the lines in the picture, how many pieces do you get?
Show the answer and working
Answer: 4 pieces
Count them one by one: 4 pieces.
Example 2
Drawing every diagonal from one corner of a regular 8-sided shape cuts it into 8 triangles.
True or false?
Show the answer and working
Answer: False
From one corner there are 2 fewer triangles than sides: 8 − 2 = 6.
Example 3
Small equilateral triangles 1 cm a side make one big equilateral triangle 4 cm a side. How many small triangles does it take?
Show the answer and working
Answer: 16 triangles
Row by row from the top: 1, 3, 5, …; 4 rows make 4 × 4 = 16.
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