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Hexagon

by goyalsau » Thu Nov 18, 2010 11:43 pm
A regular HEXAGON has a perimeter of 30 units.what is the sum of the length of all the diagonals?


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by abhishekg21 » Fri Nov 19, 2010 12:09 am
each side of the hexagon is 30/6=5 unit.
now each point in hexagon forms equilateral trangle with the center point so each of the six triangle has side=5 unit.

hexagon can have 6C2-6 diagonals. i.e =9


out of these 9 ,3 diagnols are created by just joining opposite diagnolly points.So each of these 3 diagonal's length is 10.

rest 6 diagonals forms the 3 side of a traingle having angles 30,30 and 120.since 2 sides are 5 units so the biggest size in triangle will be 5 .3 ^1/2. so such diagnolas are 6 in number.their length=30.3^1/2

so sum of length of all diagonals=30+30.3^1/2

whats the answer ?
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by shovan85 » Fri Nov 19, 2010 1:03 am
goyalsau wrote:A regular HEXAGON has a perimeter of 30 units.what is the sum of the length of all the diagonals?


No options,
Perimeter = 30 => each side = 30/6 = 5 unit

Total number of diagonals = n * (n-3)/2 = 6 * (6-3)/2 = 9

See the below diagram.

3 Red are regular diagonal (Just opp points connected). and 6 Green other diagonals.

To find the regular diagonal length. say BE=?

Length of XY = 5 units
In triangle CBX, CB = 5 units and angle CBX = 60.
Thus by using 30-60-90 triangle rule CX = 5/2 = 2.5 (Details: https://www.themathpage.com/atrig/30-60-90-triangle.htm)
Thus EY = 2.5 units.

Thus diagonal BE = 2.5 + 5 + 2.5 = 10 units

So 3 red diagonals = 3*10 = 30 units

To find the Green diagonal length say DF = ?

In triangle DAF, DA = 10 (regular diagonal of hexagon), AF = 5
thus DF = (DA^2 - AF^2) ^ (1/2) = 75 ^ (1/2) = 5 * (1.73) = 8.66

So 6 green diagonals = 8.66 * 6 = 52 (approx)

Total = 52 + 30 = 82 units
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by goyalsau » Fri Nov 19, 2010 1:38 am
You and shovan both are absolutely right, but i have some queries, I will be really great full , If you help me understand.
abhishekg21 wrote:each side of the hexagon is 30/6=5 unit.
now each point in hexagon forms equilateral trangle with the center point so each of the six triangle has side=5 unit.

hexagon can have 6C2-6 diagonals. i.e =9


out of these 9 ,3 diagnols are created by just joining opposite diagnolly points.So each of these 3 diagonal's length is 10.
Just when i measured the length of 3 diagonals that are formed by joining the opposite ends , I realize that they are equal.
How to be sure of this,
I just draw a regular pentagon and realized that all the 5 diagonals are equal.

1. question -------I want to know How to be sure of about the equal number of diagonals in a regular polygon { just imagine that a regular polygon of 15 sides , How to know that How many equal diagonals are there in it, }
abhishekg21 wrote: rest 6 diagonals forms the 3 side of a traingle having angles 30,30 and 120.since 2 sides are 5 units so the biggest size in triangle will be 5 .3 ^1/2.
I can see that the measure of triangles will be 30 , 30 , 120 ,
Question 2. But what is the relation between the sides of 30 30 120 triangle
abhishekg21 wrote:
so such diagnolas are 6 in number.their length=30.3^1/2

so sum of length of all diagonals=30+30.3^1/2

whats the answer ?
Saurabh Goyal
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by abhishekg21 » Fri Nov 19, 2010 2:19 am
regarding your first question i think we should not distinguish between the type of diagnoal saying equal or non equal.
in case if you joining 2 opposte points and the line joining them forms a triangle beneath it then you can use normal trignometry to get the length of the diagonal.
but in case if the line joning 2 opposit end points do not form a traingle beneath it then definitley there will be some other approach to caclulate the length .As in our case we used the side of equilateral triangle or we could have also used similar triangles to know the length of the diagonal.

to your second question relationship between sides of 30,30,120 triangle is 1:1: 3^1/2
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