Triangle in a hexagon
Weekly Problem 3 - 2009
What fraction of the area of this regular hexagon is the shaded triangle?
What fraction of the area of this regular hexagon is the shaded triangle?
Age
11 to 14
Challenge level
Problem
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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Call the length of one side of the hexagon $s$ and the height of the hexagon $h$:
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So the area of the shaded triangle is $$\frac{1}{2} \times s \times h$$
Now divide the hexagon into $6$ equilateral triangles:
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Each triangle has area $$\frac{1}{2}\times s \times \frac{h}{2}$$ so the area of the hexagon is $$6 \times \frac{1}{2}\times s \times \frac{h}{2}=3 \times \frac{1}{2} \times s \times h$$ or $$3 \times \text{Shaded area}$$
So the shaded area is $\frac{1}{3}$ of the area of the whole hexagon.