03312017, 10:07 AM  #1 
Join Date: Nov 2008
Location: Yukon, OK

HexVolume
ANyone have the volume of a standard hex?
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03312017, 10:11 AM  #2 
Join Date: Jun 2013

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.

03312017, 10:45 AM  #3 
Join Date: Mar 2008
Location: Dallas, TX

Re: HexVolume
Th area of a 1yard hex is (3/4)(sqrt) 2 square yards, 0r 1.06 square yards. At 4 yards high, that comes to 4.24 square yards.

03312017, 10:47 AM  #4  
Join Date: Nov 2006
Location: Heartland, U.S.A.

Re: HexVolume
Quote:
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.
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Last edited by Captain Joy; 03312017 at 11:01 AM. Reason: added comparison to hexagon and corrected an error Prince Charon pointed out 

03312017, 10:53 AM  #5  
Join Date: Dec 2012

Re: HexVolume
Quote:
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03312017, 11:03 AM  #6 
Join Date: Jul 2008

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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03312017, 11:15 AM  #7 
Join Date: Dec 2006
Location: Meifumado

Re: HexVolume
Is your hex 1 yard from cornertocorner or sidetoside?
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03312017, 11:24 AM  #8  
Join Date: Nov 2006
Location: Heartland, U.S.A.

Re: HexVolume
Quote:
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.
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Last edited by Captain Joy; 03312017 at 11:59 AM. Reason: several, until I got it right 

03312017, 11:31 AM  #9  
Join Date: Jun 2005

Re: HexVolume
Quote:
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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03312017, 11:46 AM  #10 
Join Date: Jun 2013

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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