How do you figure out square footage of a hexagon?

[haybird]

Hello All,

I am trying to figure out the square footage of my gazbo. It has 6 equal sides.
Does anybody know how this is done? I tried a google search but my head started to hurt.
Thanks

 

[mas985]

Area = 3 * sqrt(3) / 2 * (Edge Length)^2

Area ~ 2.598 * (Edge Length)^2

Edge length is also equal to the radius at a point since it is made up of 6 equilateral triangles.

 

[jschapansky]

hmm... his formula looks like it would work. but id just cut it into 6 different triangle and figure it out that way

 

[PaulR]

The formula is correct, 6 x (area of equilateral triangle) reduces to what mas985 said.

Edge length is also equal to the radius at a point since it is made up of 6 equilateral triangles.

Radius of the circumscribed circle (picky, picky). If you compare the area of that circle (pi x r^2, or ~ 3.14 x r^2) to the exact area of the hexagon (~2.6 x r^2) you see the multiplier is a little smaller for the exact area, which makes sense because the hexagon is smaller than the circle that goes around it.
--paulr

 

[gtm]

I hate to read someone else's mind, since my own is difficult enough to read, but I think mas985 was just trying to give the OP another view on how you would measure "edge length".

Although I have to say that obtaining pi as the limit of a sequence of regular polygons is a pretty cool way to present things to the students.

If only we didn't have to use that "circumscrision" word ... ouch!

Just in fun.

Cheers, Gary

 

[ckoch1969]

A hexagon is two equal trapezoids; therefore 2(base 1 + base 2)/2 * height
or in other words: add the distance of one side to the distance between two opposite corners and multiply the sum by half of the distance between the midpoint of two opposite sides

 

[haybird]

Thanks for the replies,my head still hurts and I don't even think I asked the right question.
I am thinking of making my gazbo a permanent structure. I am trying to figure out how many square(100sqft) the roof will be. Cedar shingles are quite exspensive and I am trying to figure out a price just for the roof.

Here is a picture if somebody could explain it in even lamer terms that would be great.
Thanks

 

[AV8TOR]

Here is a picture if somebody could explain it in even lamer terms that would be great.
Thanks

Give me some dimensions and I will give you the square footage.
What is the dimension from one flat wall to another, or what is the dimension from one corner to the adjacent corner (point to point)?

 

[haybird]

Thanks for you help Av8tor. Each flat wall is 6 1/2 ft. From corner to coner is 12 1/2ft.

 

[mas985]

Ah, well then what you have is a hexagonal pyramid. The surface area will be slightly more than the flat hexagon. If you want to match the exact size and shape of the umbrella, you can use direct measuremens from the umbrella.

In a hexagonal pyramid, the triangles are no longer equalateral but since you can measure it directly, the formula is much easier.

Surface Area = 6 * Triangle Area = 6 * (1 / 2 * B * H) = 3 * B * H

B = the length of one side of the hexigon
H = distance from the top of the umbrella to the mid point on a side measure on the surface (a flat hexagon reduces to sqrt(3) / 2 * B as previously shown)

However, you might want to plan for some waste so you don't come up short.

[EDIT] You must of posted while I was writing this. One more measurement would be useful if you want the exact surface area. Please measure from the top of the umbrella to the mid point on one of the sides. Otherwise, the surface area will be only the projection of the umbrella and you will come up short on material. A flat hexagon would be about 110 sq-ft.

 

[ckoch1969]

Or, half of the perimeter multiplied by the slant height ( the distance from the base of one of your triangles to the apex )

 

[AV8TOR]

Thanks for you help Av8tor. Each flat wall is 6 1/2 ft. From corner to coner is 12 1/2ft.

101.487 square feet if it is 12.5 feet corner to corner. In CAD however a hexagon 12.5 feet across points then shows the flat walls to be 6.25 feet (not 6.5). This should be close enough for you.

This is area of the flat floor

 

[mas985]

101.487 square feet if it is 12.5 feet corner to corner. In CAD however a hexagon 12.5 feet across points then shows the flat walls to be 6.25 feet (not 6.5). This should be close enough for you.

This is area of the flat floor

Unless I misread Haybird's intensions, he is trying to replace the umbrella with a roof of the same pitch so it is not flat. Depending on the pitch of the roof, the error can be quite high.

Assuming the 6.5' side length measurement is correct and a 30 degree pitch on the roof, the area would be 126.75 sq-ft.

 

[haybird]

Thanks for your help. It will give me a good idea what my price will be.