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Re: Hexagon sum of lengths of diagonals [#permalink]
30 Apr 2012, 00:27

In a regular hexagon length of each diagonal is twice of each side.Since there are 3 diagonals and 6 sides, sum of lengths of diagonals will be equal to perimeter of hexagon.You can think a hexagon as six equilateral triangles joined together.

Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink]
30 Apr 2012, 02:23

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gmihir wrote:

A regular hexagon has a perimeter of 30 units. What is the sum of the lenghts of all its diagonals ?

Sorry, I can't recall the answer choices but am sure answer 30 is incorrect one.

Look at the diagram below:

Attachment:

Hexagon.png [ 17.48 KiB | Viewed 5752 times ]

There are 9 diagonals in a hexagon.

Each of 3 red diagonals equal to \(2*side=2*5=10\) (since regular hexagon is made of 6 equilateral triangles); Each of 6 blue diagonals equal to \(2*(side*\frac{\sqrt{3}}{2})=5\sqrt{3}\) (notice that in 30°, 60°, 90° triangle, where the sides are always in the ratio \(1 : \sqrt{3}: 2\), half of the blue diagonal is the leg opposite 60°, so it equals to \(side*\frac{\sqrt{3}}{2}\));

So, the sum of the lenghts of all diagonals is \(3*10+6*5\sqrt{3}=30(1+\sqrt{3})\).

Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink]
03 May 2012, 20:53

Another good example of Deception, Excellent question and excellent explanation by bunuel. The ratio of the sides are deduced from the SINE FORMULA ie (a/sinA)=(b/sinB)=(c/sinC).

Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink]
29 Jun 2013, 21:44

Bunuel wrote:

Each of 6 blue diagonals equal to \(2*(side*\frac{\sqrt{3}}{2})=5\sqrt{3}\) (notice that in 30°, 60°, 90° triangle, where the sides are always in the ratio \(1 : \sqrt{3}: 2\), half of the blue diagonal is the leg opposite 60°, so it equals to \(side*\frac{\sqrt{3}}{2}\));

Bunuel: Can you please explain the above mentioned step in detail. How did you calculate the 90 degree angle or the small 30 degree angle ?

Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink]
28 Apr 2015, 04:48

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