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A regular hexagon has a perimeter of 30 units. What is the

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A regular hexagon has a perimeter of 30 units. What is the [#permalink] New post 30 Apr 2012, 00:02
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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.
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Re: Hexagon sum of lengths of diagonals [#permalink] New post 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.

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Answer is 30.
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Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink] New post 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
Hexagon.png [ 17.48 KiB | Viewed 4295 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}).

Hope it's clear.
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Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink] New post 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).
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Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink] New post 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 ?

Thank you
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Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink] New post 16 Oct 2013, 01:58
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Formula for no. of diagonals in a N sided polygon =

n(n-3)
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Re: A regular hexagon has a perimeter of 30 units. What is the [#permalink] New post 26 Dec 2013, 11:24
I think the right answer should be 30(1+2 squareroot3)
Re: A regular hexagon has a perimeter of 30 units. What is the   [#permalink] 26 Dec 2013, 11:24
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