How to calculate cutting angle for a circle of tiles?

I'm looking to calculate the angle I need to cut on square tiles that will be arranged around the edge of a circular pool.

Given that radius is 7' and that tile width is 12", how many tiles are needed to close the full circle, when distance between each tile is 1/4". At what angle does each tile need to be cut on both sides, so that when laid in a circle, sides of the tiles are parallel to each other? Can somebody point me in how to calculate this?

12-1/4 inches if you leave the tile full-width at the outside edge - tile plus grout.

2 pi R is circumference - 43.923 feet. 527.79 inches.

Divide by 12.25 to get number of tiles. 43.084 - depending on the project, you either figure 43 is close enough (0.27 inch grout line rather than 0.25) or you use 44 tiles and trim more off them (not full width at edge - 11.745" at outer edge.)

360 degrees in a circle divided by number of tiles = the angle of cut

360 degrees divided by 43 = 8.37 degrees per tile, to make symmetric, cut 4.185 degrees on each side. For 44 tiles, 8.18, 4.09.

To get fussy (the 43/44 may be near enough to make it matter) you could check with the fact that the edge of the tile is not really "the circumference of the circle" but a chord making up a polygon inscribed in the circle. Chord length = 2 * r * sin (c/2) where c is the "central angle" - so c/2 is 4.185 or 4.09 degrees for 43 or 44 tiles, respectively. Looks like it might make it 0.26 rather than 0.27 inch grout line for 43.

In real life, start from opposite sides and work to the point where you have one tile left on each side - check the fit of that as constructed and trim to fit as needed, since getting grout lines precisely repeatable to the 100th of an inch is unlikely in practice.

Assuming you are just going to tile a simple circle with a radius of 7', the calculation is easy to estimate number of tiles.

πR2 = Area

I'm going to use inches here: Radius is 84" = surface area of 22,156" squared.

It wouldn't be out of the question to calculate the number of tiles out of this, and add in a percentage for breakage/cuts. based on a 12" tile and 1/4" grout, I'd guess 150 tiles + 20% breakage/waste, or 177 tiles.

The next part gets more complicated. What I believe you're suggesting is cutting trapezoidal sections of tile, and arrange them in concentric rings. If you want to maximize the tile size, the pitch will be the same, but will not have the same layout on each and every tile. For each consecutive ring of tiles form the center, they will become more oblique and large. I suggest doing an accurate scaled drawing in the spirit of what I threw together. Using autocad or sketchup would be highly advisable to make a 2D drawing. This is really the easiest way in my mind. Then you can have a firm plan of cuts, measurable angles, as well as a count of tiles.

Keep in mind, this is just about layout and measurements. I have NO idea of the tiles cut like this will be stable or if it will be easy to install.

Take your time, measure twice, cut once and when in doubt, step away and ask yourself "Is this work going to be worth the outcome".

Good luck and let us know how it goes!

I accepted answer but wanted to provide my line of thinking also, and some facts that were missing in initial question, and that is that length of the tile is 18" and that there will be a 1" overhand over the circle. Here is how I calculated it:

Circumference of the circle is 43.98, let's round it up to 44. Since I want the tiles at the end not to be cut, and there will be overhand of 1", outer circle radius is 84 + 17 = 101. So outer circle circumference is 52.88', lets round it up to 53'. So basically inner circle is 9 feet shorter then outer one. Inner circle due to gap of 1/4" between tiles can hold only 43 tiles. So from these 43 tiles I have to take away 9'. 9x12 = 108/43 = 2.51. So from each tile I have to take away 2.5 inches, so on each side that is 1.25". Now if we apply trigonometry, since we know all the pieces we can calculate the angle of the cut. Based on opposite and adjacent side length, angle on the top of the tile is 3.97 degrees. I think that is pretty close to what others came up with. It is not absolutely necessary that it fits exactly as the circle will not be closed, only half circle will be created. This is for the pool coping.