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

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New post 30 Apr 2012, 01:02
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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: A regular hexagon has a perimeter of 30 units. What is the  [#permalink]

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New post 30 Apr 2012, 03:23
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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 14542 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: Hexagon sum of lengths of diagonals  [#permalink]

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New post 30 Apr 2012, 01:27
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.

Hope it helps.. :)
Answer is 30.
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Re: A regular hexagon has a perimeter of 30 units. What is the  [#permalink]

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New post 03 May 2012, 21:53
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]

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New post 29 Jun 2013, 22:44
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]

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New post 16 Oct 2013, 02:58
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Formula for no. of diagonals in a N sided polygon =

n(n-3)
--------
2
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Re: A regular hexagon has a perimeter of 30 units. What is the  [#permalink]

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New post 26 Dec 2013, 12:24
I think the right answer should be 30(1+2 squareroot3)
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Re: A regular hexagon has a perimeter of 30 units. What is the  [#permalink]

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New post 20 Apr 2017, 05:51
PareshGmat wrote:
Formula for no. of diagonals in a N sided polygon =

n(n-3)
--------
2



Just to clarify:

you mean n(n-3) divided by 2 right?
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Re: A regular hexagon has a perimeter of 30 units. What is the  [#permalink]

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New post 21 Apr 2017, 09:44
I have a answer but can someone find any flaw in it? Becoz Im not sure if its right or not...


S= side of hexagon
D = diagonal of a regular hexagon

6*S = 30

S=5 ;

Property: The diagonal of a regular hexagon, is twice the side length.

D = 2*S
D = 10;

Regular hexagon has 6 diagonals

6*D is sum of all diagonals

60 Answer
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Re: A regular hexagon has a perimeter of 30 units. What is the  [#permalink]

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New post 21 Apr 2017, 09:45
Bunuel wrote:
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

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.





I have a answer but can someone find any flaw in it? Becoz Im not sure if its right or not...


S= side of hexagon
D = diagonal of a regular hexagon

6*S = 30

S=5 ;

Property: The diagonal of a regular hexagon, is twice the side length.

D = 2*S
D = 10;

Regular hexagon has 6 diagonals

6*D is sum of all diagonals

60 Answer
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Re: A regular hexagon has a perimeter of 30 units. What is the  [#permalink]

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Re: A regular hexagon has a perimeter of 30 units. What is the   [#permalink] 28 Nov 2018, 00:05
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