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Area of a hexagon..
By Louise on Monday, August 06, 2001 - 11:13 am:

What is the area of a hexagon with respect to one of its six sides? Can someone prove this for me?
Thanks Louise

By Susan Wakeling on Monday, August 06, 2001 - 03:04 pm:

I assume we are dealing with a regular hexagon (i.e. all sides the same length etc). Divide the hexagon into six triangles, so the central corner must have an angle of 60 degrees, and the other two angles must be the same as each other, and the total must be 180. So all angles of the triangle are 60 degrees. Can you work out how large the triangle is now? Then multiply it by 6 for the size of the hexagon.

By Louise on Tuesday, August 07, 2001 - 03:30 pm:

Take a regular hexagon with its six sides of length a. The 6 equilateral triangles contained within this hexagon also have all their sides equal to a.
The area of a triangle is its base (in this case a)x by its perpendicular height (which is unknown). I thought that if i could find the perpendicular height in terms of a by manipulating Pythagoras' theorem then I would be able to determine the area of the hexagon with respect to a.
Do you see what I'm rambling on about?
By using Pythagoras: let the perpendicular height be p so;
a2=a2+p2 as sides of a equilateral triangle have equal length don't they...
p=root(a2-a2) which doesn't make sense...
Where am I going wrong?
Louise x

By Susan Wakeling on Tuesday, August 07, 2001 - 03:56 pm:

I assume that you are considering the triangle running from half way along one side to the opposite corner. So sides have length a, a/2 and p. Not a,a and p! So it is a2=(a/2)2+p2.
Can you take it from there?

By Arun Iyer on Tuesday, August 07, 2001 - 08:13 pm:

In a more simple way,
angle of a equilateral triangle is 60.
so sin 60 = p/a
hence p = sqrt(3) × a/2

area of the equilateral triangle
= 1/2 × p × a
= 1/2 × sqrt(3) × a/2 × a
= sqrt(3) × a2/4

hence carry on from here..
love arun

By Louise on Thursday, August 09, 2001 - 01:27 pm:

Doh!
Louise x

By Arun Iyer on Thursday, August 09, 2001 - 06:57 pm:

Well louise,
What does "Doh!" mean?

love arun

By Louise on Sunday, August 12, 2001 - 04:31 pm:

Do u not watch the simpsons?
Louise x aka Homer

[For other international readers, The Simpsons is a cartoon, very popular in the USA and UK. Louise's "Doh!" was an expression of annoyance with her own error! - The Editor]