# Tag Archives: Five Triangle Puzzle ### The Five Triangle Puzzle: Solutions

The Five Triangle Puzzle was the subject of a post back on February 11. I’m hopeful that some of you will have downloaded the pieces and solved the puzzle. The challenge was to put all five pieces together to form a triangle and then to determine if there were other triangles that could be formed using all five pieces.

I’m guessing that you probably used a trial and error strategy to find the solutions, which was my strategy the first time I tried it. What happened to me was that I first found a way to make an equilateral triangle. This came about because I fiddled around with how to match the long leg of the larger triangle with the hypotenuse of the smaller triangle. I noticed that if I formed an equilateral triangle with the two smaller triangles, then that triangle had three sides the length of the long leg to the larger triangle. From that point on, it is simply a matter of arranging the large triangles around the two smaller ones. At each vertex of the triangle formed is the 60-degree angle of one of the larger triangles.

Once I discovered this solution to the puzzle, I never forgot it. If you give me the five triangles, I can immediately put them together to form this next triangle. The second triangle that can be formed is, at least for me, a bit more of a challenge. But for the life of me, I have no clue as to how to proceed–even after having done it dozens of times. Even though I remember that this solution has one angle of 120 degrees and two angles of 30 degrees, I still have trouble figuring out how to put them together. These are the only two ways to put these pieces together to form a triangle. Did anyone try for a 30-60-90 triangle? That would seem like a reasonable possibility, right? In fact, a 30-60-90 triangle can’t be made with these five triangles. For those who might be interested, I’ll present in a follow-up post a fairly simple argument for why such a triangle is not possible. ### The Five Triangle Puzzle: Composing and Decomposing

Beginning in first grade and continuing through the grades, the Common Core Math Standards emphasize composing and decomposing shapes. One way to give students experience with composing shapes is through put-together puzzles. Actually, working at put-together puzzles involves lots of composing and decomposing. Sometimes what we compose is a solution; sometimes it’s not. While the… Continue Reading