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Geometry Level pending

If a regular hexagon and a regular Dodecagon (12 sided shape) have the same side lengths what is the (fully simplified) ratio of their areas? The answer is in the form:
Area of hexagon : Area of Dodecagon

\sqrt {2} : 1+\sqrt {2}

2 \sqrt {2} : 3+3\sqrt {2}

\sqrt {3}: 3+2\sqrt {3}

\sqrt {3} : 4+2 \sqrt {3}

1 solution

Curtis Clement
Nov 22, 2014

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Draw squares on each side of a hexagon and join the vertices (as shown). It is well known that the interior angle in a hexagon is 120 degrees. The two right angles sum to 180 degrees, so angle(BAC)=60 degrees. Also, AB=AC (congruent squares), so triangle ABC is equilateral. By special triangles, the height of the equilateral triangles (and half the hexagon) is $0.5x\sqrt{3}$ . Also, the diagonal of the hexagon is 2 ${x}$ . Hence, the area of the hexagon is (1 ) $1.5x^{2}$ $1\sqrt{3}$ , the area of the squares is $6x^{2}$ and the area of the equilateral triangles is equal to the area of the hexagon (by reflection, if you split up the hexagon into 6 equilateral triangles). It follows that the total area of the dodecagon is $3x^{2}$ *[ $\sqrt{3}$ +2]. Hence, the required ratio is $1\sqrt{3}$ : 4 + $2\sqrt{3}$

Curtis, to get your Latex expressions to display, you have to place brackets around them, as in $\backslash ( \text{Latex code } \backslash )$ . I'ved edited part of your solution to give you a reference.

Otherwise, your latex expressions look good.

Note: The same also applies to the MCQ options.

Calvin Lin Staff - 6 years, 6 months ago

Sorry, my bad. I've cleaned up my solution now :)

Curtis Clement - 6 years, 6 months ago

Thanks, it looks good now.

Isn't it exciting to be able to use Latex?

Calvin Lin Staff - 6 years, 6 months ago

Note: For the hexagon, the height of the equilateral triangle is $\frac{ x \sqrt{3} } { 2}$ , instead of $x \sqrt{3}$ . As such, the area of the hexagon is $\frac{ 3 \sqrt{3}} { 2} x^2$ instead.

This error carries over to finding the area of the dodecagon, which should be $3 ( 2 + \sqrt{3} ) x^2$ .

Hence, the ratio is $\sqrt{3} : 4 + 2 \sqrt{3}$ .

I have updated the answer accordingly.

Calvin Lin Staff - 6 years, 6 months ago

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