Horrible Hexagons #3

Geometry Level 2

The diagram above shows a regular hexagon ${ H }_{3 }$ with area $H$ which has six right triangles inscribed in it. Let the area of the shaded region be $S$ , then what is the ratio $H:S?$

7:2

4:1

9:2

5:1

16:3

6:1

1 solution

Nguyen Thanh Long
Apr 9, 2015

We can consider the large hexagon as 6 equilateral triangles, of which 2 of them have a shaded part. For each triangle, the amount that is shaded is $\frac{2}{3}$ . Hence, the shaded ratio is

$\frac{2}{6}*\frac{2}{3}=\frac{2}{9}$

Is this solution correct. I thought it was 7:2

Adam Westwood - 5 years, 6 months ago

The ratio of the whol are to the shaded area is 9:2, but the ratio of the unshaded to shaded areas is 7:2

Adam Westwood - 4 years, 10 months ago

9:2 is the ratio between H_3:S. But if the right-angled triangles have each area H, then H:S=6:4=3:2

JnPau BrHi - 5 years, 6 months ago

Actually, H_3 is hexagon, but its area is H, whereas S is the shaded area. So:

$"H : S"$ $"Hexagon Area : Shaded Area"$

Consider 6 equilater regions with 6 mini rectangle triangles each. So you'll get 36 mini triangles in total from which only 8 are shaded. Formula again:

$36:8 = 9:2$

Javier Francés - 5 years ago

How do you know that the shaded portion of each triangle is 2/3, is that an approximate, and is there a way to find this ratio for sure?

Manuela Bhananker - 4 years, 1 month ago

I maintain that the ratio is 4 :1. And there are 8 shaded right triangles, not 6.

Brody Burkett - 3 years ago

Im pretty sure the solution is 7:2

Joe Chesney - 5 years ago

The hexagon is composed of 6 isoceles triangles. In the sketch above, you just have to draw their apothem and then you get 6x2 right-angled triangles. So you have the same little right-angled triangles composing the whole hexagone. Each isoceles triangle has 6 little right-angled triangles and the total is LaTex: $\color{#3D99F6}{6_{\text{isoceles} \triangle} \times 6_{\text{little right-angled triangle}}} = 36.$ You count LaTex: $\color{#3D99F6}{4_{\text{little right-angled} \triangle} \times 2_{\text{shaded area}} = 8_{ \text{ shaded little right-angled triangles}}}.$ So, the ratio is LaTex: $\frac{36}{8} = \color{#D61F06}{\frac{9 \times 4}{2 \times 4} = \boxed{\frac{9}{2}}}$

Frédéric Deleria - 4 years, 1 month ago

Horrible Hexagons #3

1 solution

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