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?
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![Triangle in a Hexagon Triangle in a Hexagon](/sites/default/files/styles/large/public/thumbnails/content-id-6232-Weekly%2525202009%252520-%2525203%252520Problem.png?itok=prIijpN1)
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Call the length of one side of the hexagon $s$ and the height of the hexagon $h$:
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![Triangle in a Hexagon Triangle in a Hexagon](/sites/default/files/styles/large/public/thumbnails/content-id-6232-solution1.jpg?itok=1zCEudvM)
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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![Triangle in a Hexagon Triangle in a Hexagon](/sites/default/files/styles/large/public/thumbnails/content-id-6232-solution2.jpg?itok=GGIBfIAJ)
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.