Three whole-number sides that add up to twenty — how many different triangles can you make? Fewer than you would guess, because most ways of splitting twenty into three parts cannot bend into a triangle at all.
Order the sides so the shortest comes first: a, then b, then c, adding to the perimeter. Three lengths only close into a triangle when the two shorter ones out-reach the longest — a plus b has to be more than c — otherwise they lie flat and never meet above it. That single condition, the triangle inequality, is the whole puzzle: split the perimeter every way you can, then keep only the splits that pass it.
Why this sheet works
The mistake almost everyone makes is to count every way to write the perimeter as three whole parts and stop there. For a perimeter of twenty that is far too many, because splits like 1, 1, 18 or 2, 3, 15 are on the list and none of them is a triangle — the short sides cannot span the long one. Seeing which splits to throw away is most of the work, and it is the thing the sheet is really teaching.
Count the triangles with whole-number sides and the given perimeter, and show why the impossible ones are out.
119 matchsticks of equal length are laid end to end to make the three sides of a triangle, with none left over and none broken.How many different triangles can be made? (The same triangle turned around counts once.)Why
2A farmer has 27 metres of fence and wants a triangular pen whose three sides are each a whole number of metres.How many differently shaped triangular pens can be built? (Turning or flipping a pen over does not make it a new one.)Why
Integer trianglesPerimeter up to 30
Name: Date:
Count the triangles with whole-number sides and the given perimeter, and show why the impossible ones are out.
329 matchsticks of equal length are laid end to end to make the three sides of a triangle, with none left over and none broken.How many different triangles can be made? (The same triangle turned around counts once.)Why
4A farmer has 21 metres of fence and wants a triangular pen whose three sides are each a whole number of metres.How many differently shaped triangular pens can be built? (Turning or flipping a pen over does not make it a new one.)Why
Integer triangles · Perimeter up to 30Answer key
1. Sides a ≤ b ≤ c with a + b + c = 19 only make a triangle when a + b > c — the two shorter sides have to out-reach the longest, or they never close up. So a split like (1, 1, 17) is out. Walking the shortest side up from 1 and keeping the splits that obey it: (1, 9, 9), (2, 8, 9), (3, 7, 9), (3, 8, 8), (4, 6, 9), …, (6, 6, 7). That is 10 different triangles.2. Sides a ≤ b ≤ c with a + b + c = 27 only make a pen when a + b > c — the two shorter sides have to out-reach the longest, or they never close up. So a split like (1, 1, 25) is out. Walking the shortest side up from 1 and keeping the splits that obey it: (1, 13, 13), (2, 12, 13), (3, 11, 13), (3, 12, 12), (4, 10, 13), …, (9, 9, 9). That is 19 different triangles.3. Sides a ≤ b ≤ c with a + b + c = 29 only make a triangle when a + b > c — the two shorter sides have to out-reach the longest, or they never close up. So a split like (1, 1, 27) is out. Walking the shortest side up from 1 and keeping the splits that obey it: (1, 14, 14), (2, 13, 14), (3, 12, 14), (3, 13, 13), (4, 11, 14), …, (9, 10, 10). That is 21 different triangles.4. Sides a ≤ b ≤ c with a + b + c = 21 only make a pen when a + b > c — the two shorter sides have to out-reach the longest, or they never close up. So a split like (1, 1, 19) is out. Walking the shortest side up from 1 and keeping the splits that obey it: (1, 10, 10), (2, 9, 10), (3, 8, 10), (3, 9, 9), (4, 7, 10), …, (7, 7, 7). That is 12 different triangles.
