Layout geometry
Diagonal (45°) layout: tile diagonal and edge half-tiles
- where
- tile diagonal (mm)
- square tile side (mm)
- diagonal pitch along the wall (mm)
- joint width (mm)
- half-tile triangles along the run
- wall run length (m)
- whole tiles cut to make the triangles
Explain this equation
Gives the key sizes for laying square tiles at 45°: the corner-to-corner size of a tile, how far the pattern steps along a wall, and how many half-tile triangles you need along an edge.
d is the diagonal of a tile of side L. p is the pitch along the wall: the diagonal of a tile plus its joint, because each diagonal row repeats every tile and joint. R is the wall run in metres, converted to millimetres by the 1000. Nt is the number of edge triangles and Nw the whole tiles you cut in half to make them.
By Pythagoras the diagonal of a square is √2 times its side, about 1.414 times. Dividing the run by the pitch and rounding up gives the triangles, and each tile cut corner to corner gives two triangles, hence Nt divided by 2.
Sense check: a 300 mm tile is about 424 mm corner to corner, and with a 3 mm joint the pattern steps about 428.5 mm along the wall, so a 3.6 m wall needs 9 edge pieces.
Watch out: the end pieces are only exact halves if the run is a whole number of pitches. Set out from the centre so the ends match.
Geometry for square tiles laid at 45°: the corner-to-corner size, how far each tile steps along a wall, and how many half-tile triangles you need along one edge.
Enter your values to see the result.
Good to know
A square tile cut corner to corner gives two right-angled triangles whose long side is the tile diagonal √2 × L. Along a wall the diagonal grid repeats every √2 × (L + j). Run this for each wall to get the edge triangles; the four room corners need quarter pieces. The total tile count for the floor is the same as for a straight lay, plus the higher waste allowance (about 15%).
Common mistake
Using the tile side instead of the diagonal pitch when gauging along the wall, so the cut pieces drift out of balance by the end of the run.
Limits
Only exact halves fit where the grid lines meet the wall at a tile corner. Set out from the centre so the edge pieces are equal; the end pieces of a run will rarely be exact halves unless the run is a multiple of the pitch.
Sources
Results are calculations from the published methods and manufacturer figures shown, and depend on the data you enter. Always follow the tile, adhesive and grout manufacturers’ instructions, BS 5385 and the Tile Association guidance. Coverage rates vary by product and substrate. Check prices and VAT rules with your own records and HMRC.