Tiling a square with right-angled triangles

Question

Tile a square with twenty congruent right-angled triangles. For each triangle, one leg is of length 1 and the other leg is of length 2.

Gareth McCaughan · Accepted Answer · 2024-02-01T01:40:45.690

28

Note first that

the area of the square is 20 times the area of the triangle, namely 20, so the side length is $2\sqrt5$.

That's

twice the hypotenuse of the triangle

which immediately suggests the overall shape of the thing. With apologies for the horrific ASCII art:

.   .   .   .   .   .   .
            ..''|:
.   .   ..''.___. : .   .
    ..''|       |  :
..''.___.___.___.___:   .
 :  |       |       |:
. : .___.___.___.___. : .
   :|       |       |  :
.   :___.___.___.___.___:   .   .
     :  |       |   ..''`
.   . : .___.___..''.   .   .   .
       :|   ..''
.   .   :.''.   .   .   .   .   .

where

each of those 2x1 rectangles is of course divided into two triangles by one of its diagonals.

Here, have some slightly less horrific not-ASCII not-art:

edited Feb 01 '24 at 01:40

answered Feb 01 '24 at 01:29

Gareth McCaughan

119,120
7
313
454

3

I like this answer because it explains the math involved plus it gives two views of a solution in two different ways: ASCII art vs Graphic image AND Tilted vs Non-tilted square. I also appreciate the guide dots and the guidelines. – Will Octagon Gibson Feb 01 '24 at 03:42
1

That ascii art is fantastic! – LeppyR64 Feb 01 '24 at 12:58
1

I upvoted this just for the ASCII art :) – Dmitry Kamenetsky Feb 01 '24 at 13:03

score 24 · Answer 2 · answered Feb 01 '24 at 01:33

24

May I offer a more aesthetic tiling?

answered Feb 01 '24 at 01:33

Daniel Mathias

14,926
2
30
62

1

More aesthetic than ASCII at you mean? :) – M Oehm Feb 01 '24 at 05:27
@MOehm Certainly - Gareth added the image about seven minutes after I posted. Even so, I find the mirror symmetry more pleasing to the eye. – Daniel Mathias Feb 01 '24 at 05:48
9

Which Transformer is this? – codewarrior0 Feb 01 '24 at 10:27
reminds me a bit of a rot13(fgne qrfgeblre) shape... (saw that as the first after unspoilering the solution) – masterX244 Feb 01 '24 at 18:43

score 17 · Answer 3 · answered Feb 01 '24 at 11:21

Using the solution of this classic puzzle, "If the dotted lines connect the midpoint of each edge with a vertex, find the fraction of the area of the yellow square to the large red one":

namely,

1/5 i.e. the same as each of the 4 red triangles.

Each red triangle and the yellow square can be divided into 4 of the triangles given in the question, resulting in this neat solution:

Tiling a square with right-angled triangles

3 Answers3