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Old 03-31-2017, 10:07 AM   #1
Refplace
 
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Default HexVolume

ANyone have the volume of a standard hex?
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Old 03-31-2017, 10:11 AM   #2
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Default Re: HexVolume

A standard hex is one yard wide and 4 yards tall. Assuming a circular base, it ends up being exactly pi cubic yards in volume.
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Old 03-31-2017, 10:45 AM   #3
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Default Re: HexVolume

Th area of a 1-yard hex is (3/4)(sqrt) 2 square yards, 0r 1.06 square yards. At 4 yards high, that comes to 4.24 square yards.
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Old 03-31-2017, 10:47 AM   #4
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Default Re: HexVolume

Quote:
Originally Posted by Varyon View Post
A standard hex is one yard wide and 4 yards tall. Assuming a circular base, it ends up being exactly pi cubic yards in volume.
This wasn't immediate obvious to me, so I thought I'd run the numbers for you doubters.

Area of a cylinder = pi•r^2 • height

In our case, the radius, r = ½y
and height = 4y
where y = 1 yard

So, pi•(y/2)^2•4y = pi • (y^2 / 4) • 4y = pi • y^3 • 4 / 4 = pi•y^3
= 3.14 cubic yards

I'm surprised it's so much less than the hexagon value.

Last edited by Captain Joy; 03-31-2017 at 11:01 AM. Reason: added comparison to hexagon and corrected an error Prince Charon pointed out
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Old 03-31-2017, 10:53 AM   #5
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Default Re: HexVolume

Quote:
Originally Posted by Captain Joy View Post
This wasn't immediate obvious to me, so I thought I'd run the numbers for you doubters.

Area of a cylinder = pi•r^2 • height

In our case, the radius, r = ½y
and height = 4y
where y = 1 yard

So, pi•(y/2)^2•4y = pi • (y^2 / 4) • 4y = pi • y^3 • 4 / 4 = pi•y^3
= 3.14 square yards

I'm surprised it's so much less than the hexagon value.
Did you mean cubic yards, there?
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Old 03-31-2017, 11:03 AM   #6
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Default Re: HexVolume

I get a different value for the hexagon area. 3*1/(2 sqrt(3))*1 square yard, leading to a 3.464 cubic yard volume.
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Old 03-31-2017, 11:15 AM   #7
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Default Re: HexVolume

Quote:
Originally Posted by Ulzgoroth View Post
I get a different value for the hexagon area. 3*1/(2 sqrt(3))*1 square yard, leading to a 3.464 cubic yard volume.
Is your hex 1 yard from corner-to-corner or side-to-side?
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Old 03-31-2017, 11:24 AM   #8
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Default Re: HexVolume

Quote:
Originally Posted by Ulzgoroth View Post
I get a different value for the hexagon area. 3*1/(2 sqrt(3))*1 square yard, leading to a 3.464 cubic yard volume.
A hex is basically six equilateral triangles. The area of a triangle is ½bh. A GURPS hex is 1 yard wide, from side to side.

Set y = 1 yard.
This means the area of each triangle, ½bh = ½ • (1/SQRT(3))y • (½)y
= (1/(4•SQRT(3)))•y^2.
Since six of these triangles make a hex, hex area = (6/(4•SQRT(3)))y^2.
The volume is 4y high, so hex volume = 4y•(6/(4•SQRT(3)))y^2 = (6/SQRT(3))•y^3
= 3.46 cubic yards, in agreement with Bill Stoddard, which pleases me no end.

Last edited by Captain Joy; 03-31-2017 at 11:59 AM. Reason: several, until I got it right
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Old 03-31-2017, 11:31 AM   #9
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Default Re: HexVolume

Quote:
Originally Posted by Varyon View Post
A standard hex is one yard wide and 4 yards tall. Assuming a circular base, it ends up being exactly pi cubic yards in volume.
A hex does not have a circular base, but a hexagonal one.

Assume a distance of 1 yard from hex center to hex center. Then the distance from hex center to the middle of a given edge is 0.5 yard. We can divide the hex into six equilateral triangles. Taking the line from the hex center to the middle of the edge as the adjacent side, and applying a bit of trigonometry, we get that the line from the center to the vertex is 0.57735 yards, and therefore any exterior side must be the same length, because the triangle is equilateral. Its area is therefore 0.1443 square yards, and the area of the entire hex is 0.866 square yards, or just about 13/15 square yards. Since the height is 4 yards, the volume is 3.464 cubic yards.

For comparison, an inscribed circle has radius 0.5 yards and area 0.785 square yards, and a circumscribed circle has radius 0.57735 yards and area 1.047 square yards. The former gives volume equal to pi; the latter gives volume equal to 4.1888 cubic yards.
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Old 03-31-2017, 11:46 AM   #10
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Default Re: HexVolume

GURPS hexes are, to my knowledge, meant to be approximations of circles. What the actual volume works out to depends on which side of "circle" you err - if you treat a GURPS hex as being the largest hexagon that will fit within a 1 yard diameter circle (the points of the hexagon touch the circle), you'll end up with volume less than pi. If you treat a GURPS hex as being the smallest hexagon that a 1 yard diameter circle will fit inside of (the edges of the hexagon tough the circle), you'll end up with a volume greater than pi. With the way GURPS hexes are arranged - going from the center of one hex to the center of another (a nominal yard) passes through an edge, not a point - I believe the latter interpretation is more correct than the former, so you'd expect a volume greater than pi. That's only if you treat the approximation as literal truth, but in that case movement is going look a little odd.

EDIT: Partially ninja'ed by Bill. I feel my point about it being an approximation of a circle stands.
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