Cut up and rearrange puzzles


Problem: What fraction is shaded?


Solution: There are two easy ways to break up this shape. The first, is to cut up the figure into congruent equilateral triangles. Focusing on this, you will find 6 of the 18 equilateral triangles are shaded, and thus, 1/3 is shaded. The other way is to cut up the figure into congruent isosceles triangles. Focusing on this, you will find 6 of the 18 isosceles triangles are shaded, so of course, just as before, 1/3 is shaded.


Problem: What fraction is shaded?


Solution: Imagine rotating the middle square 45 degrees. Now we can imagine cutting up the square into congruent isosceles triangles. There are 16 such triangles, 4 of which are shaded. Thus, 1/4 of the figure is shaded.


Problem: What fraction is shaded?


Solution 1: Cut up the figure into congruent equilateral triangles, as pictured below. You will find there are 30 blue triangles, and 24 orange triangles. Thus, 30/54 = 5/9 of the figure is blue.


Solution 2: Cut up the figure into congruent isosceles triangles, as pictured below. You will find there are 30 blue triangles, and 24 orange triangles. Thus, 30/54 = 5/9 of the figure is blue.