perimeter of a hexagon

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george coon

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I am trying to establish the length (perimeter) of a hexagon given its area. I realize a hexagon may be broken into 6 triangle and 12 30-60-90 triangles. So I have the area of the triangle and that can not change. How do I determine the length of the short leg of that triangle (X12 would give me the perimeter)? Thanks
 
george coon said:
I am trying to establish the length (perimeter) of a hexagon given its area. I realize a hexagon may be broken into 6 triangle and 12 30-60-90 triangles. So I have the area of the triangle and that can not change. How do I determine the length of the short leg of that triangle (X12 would give me the perimeter)? Thanks
Click to expand...
Is it a regular hexagon? If it is then it can be divided into six equilateral triangles and you can continue from there....
 
Subhotosh Khan said:
Is it a regular hexagon? If it is then it can be divided into six equilateral triangles and you can continue from there....
Click to expand...

No, I can't continue from there, that's the question. Also I didn't realize that a regular hexagon had equilateral triangles. I know that the area of the triangle is 1/2 the base X the height (essentially converting it to a rectangle) but how to reverse that to get a leg length when the area is known? (specifically the short leg).? I'm sure it's simple but I need help. Thanks
 
Say the area of the regular hexagon is 16 square inches, then one equilateral triangle is 2.666....sq.In., what is the length of the base? (or how to find it).
 
george coon said:
Say the area of the regular hexagon is 16 square inches, then one equilateral triangle is 2.666....sq.In., what is the length of the base? (or how to find it).
Click to expand...

The side of an equilateral triangle s=32h\displaystyle s=\dfrac{\sqrt3}{2}h where h\displaystyle h is the height.
 
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I don't understand the math, but given those formulas, the perimeter of a regular hexagon with area of 16 (square inches) is 9.114. The perimeter of a square of the same area (16 Sq In) is 16 inches. This following concept may be a leap but, the most efficient shape with the least perimeter connected to all others, is a hexagon. Bees discovered this as did soap bubbles in your bath.
( I was just thinking about bees today).
George
 
george coon said:
I don't understand the math, but given those formulas, the perimeter of a regular hexagon with area of 16 (square inches) is 9.114. The perimeter of a square of the same area (16 Sq In) is 16 inches. This following concept may be a leap but, the most efficient shape with the least perimeter connected to all others, is a hexagon. Bees discovered this as did soap bubbles in your bath.
( I was just thinking about bees today).
George
Click to expand...

You are correct.

However do you know why the most efficient structure is not an octagon (8 sided figure) or a dodecagon (12 sided figure)?
 
revision

I don't know why an octagon or a decahedron are less efficient because I would actually assume they are MORE efficient given that they approach a circle which would be the most efficient. I realize that the hexagon is the last regular polygon that be tessellated. The circle , the most efficient, can not be tessellated. A triangle can also be tessellated but I think the perimeter of a triangle with given area is more than the perimeter of a hexagon of the same area.

Hexagons.GIF
 
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george coon said:
I don't know why an octagon or a decahedron are less efficient because I would actually assume they are MORE efficient given that they approach a circle which would be the most efficient. I realize that the hexagon is the last regular polygon that be tessellated. The circle , the most efficient, can not be tessellated. A triangle can also be tessellated but I think the perimeter of a triangle with given area is more that the perimeter of a hexagon of the same area.

Hexagons.GIF
Click to expand...

You got it!!
 
Thanks for all your help Subhotosh. I also learned this in Architectural school when, as I was studying, (a very long tome ago!) I noticed the suds in my bath formed hexagons naturally and I wondered why. For some reason, the bees have this figured out! (As did the basalts in the Giants' Causeway).
 
In my quest for the perimeter of a hexagon, I discovered several things, one of which remains an added question. That is...why is the area of a equilateral triangle s = length of a side
area%20equilateral%20triangle%20formula.gif
equilateral%20triangle.gif

I understand why the area of a triangle is 1/2bXh but this assumes you know the h. Is there a simple explanation for this strange formula?
 
george coon said:
In my quest for the perimeter of a hexagon, I discovered several things, one of which remains an added question. That is...why is the area of a equilateral triangle s = length of a side
area%20equilateral%20triangle%20formula.gif
equilateral%20triangle.gif

I understand why the area of a triangle is 1/2bXh but this assumes you know the h. Is there a simple explanation for this strange formula?
Click to expand...

Yes.. But you need to remember certain geometrical facts:

1) If we drop a perpendicular from the vertex to the base (the height or h), for an isosceles triangle, the perpendicular bisects the base.

2) Call Pythagoras and prove that h = s*√3/2

3) Now apply the well known formula for area of the triangle → A = 1/2 * b * h (where b = s)

By the way use * for multiplication (standard now) - to avoid confusion with variable name x.
 
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