Practical math question: I’m trying to determine how to estimate grout coverage for a grid of hexagonal tiles. I have a tool to calculate the amount of grout required to fill the joints between square tiles, given the tile dimensions, grout joint width, and square area to grout. I’m looking for a magic ratio to adapt this coverage from square tiles to hexagonal tiles of the same diameter (not edge length). Tech services at the grout manufacturer couldn’t help, so I’m turning to the great minds of the interweb. Any ideas?
Are you doing such a large number of tiles that it’s going to make a big difference? The square tiles, if I’m thinking about this right, will use more grout. If you find out how much grout you’ll need to do the project with square tiles, you’ll be able to return the unused portion. Of course, I think, even though the tiles are the same “diameter” you’re going to need more actual tiles since they’re going to nest together differently (even thought they’re the same diameter, they’re going to be smaller) so that might throw my theory out the window.
You could give this a shot…
Can you clarify what you mean by “diameter” ? Corner to corner of the hexagon? flat to opposite flat? Same thing for the square, corner to corner or flat to flat?
Thanks for the link, Joey. Running some numbers through the calculator gave me a ratio of 1.25 (Hexagonal tiles use 25% more grout than square tiles of the same tile width).
The consensus around the shop is that that figure seems a little low. Anyone interested in putting some geometry and math into the problem is welcome to have at 'er.
@Yamato: by diameter I meant tile width: flat edge to flat edge. Hence a 2" square tile has 2" long edges, a 2" hex tile has …uhhh… 1" long edges? (Yikes, I’m rusty!)
A regular hexagonal tile’s width, defined by that link as the length from edge to edge, is the side length times the square root of 3[sup]li[/sup]. So side length is the width divided by the square root of 3. Assuming a tile width of 12 inches, the length is a shade under 7 inches (6.928). Times 6, that’s 41.57 inches in perimeter.[/li]
If the hexagon tile’s width is defined as corner to corner, a 12 inch wide tile is 36 inches in perimeter.
A square’s perimeter is just the width times 4, so a width of 12 is 48 inches. So either way you should be using less grout per tile, unless there’s some factor I’m totally not accounting for.
Less grout per tile (hypotenuse is shorter then the sum of the legs), but more tiles will fit in the same area. I mentioned this above, but I wasn’t sure which side would win out or if it would just be a wash. If Genomatic’s numbers are correct, the hex tiles use 25% more grout due to needing more tiles.
Ah, right. Good point. :smack:
Well, I’ve worked on this at home for a bit with time and Wolfram at my disposal, and I still haven’t nailed it. The approach I’ve taken is to determine the area to be grouted by subtracting the area occupied by tiles from the total area being tiled.
Ignoring the edges, this is the best I can make of it:
let
A = total area being tiled
W = width of tile (between parallel edges)
J = grout joint space
The formula for a hexagon’s area in terms of W is (root 3 / 2) W^2.
Grout area = A - Tile Area
for square tiles:
Tile Area = W^2 x Number of tiles
Number of tiles = A / (W + J)^2
for hex tiles:
Tile Area = (root 3 / 2) W^2 x Number of Tiles
Number of tiles = A / ( (root 3 / 2) (W + J)^2 )
Now i figure i’ve mucked something up along the way because when I multiply the number of tiles by the area of a single tile the constant cancels out and I end up with the same formula for both square and hex. And that can’t be right.
The other approach would be to determine the number and length of the grout lines, then calculate the total area and subtract the area of 2/3 of the intersections.
Come to think of it, I think that’s correct. Consider: Draw a line down the center of each grout-line, and then draw lines from the center of each tile to all of that tile’s corners. This divides the floor up into triangles, each of which has base (with half a grout-line on it) equal to the side length of a tile, and height equal to half the width of a tile. Call the base length b[sub]h[/sub] for the hexagon tiles and b[sub]s[/sub] for the square tiles, the height h, and half the thickness of the grout lines t. Then each triangle has grout-area bt and tile-area 1/2 hb, so the ratio of tile-area to grout-area is h/2t. b (whether it was [sub]h[/sub] or [sub]s[/sub]) canceled out, and h and t are the same for both shapes, so you’re always going to need the same proportion of your floor taken up by grout, no matter what tile shape you use.