Problem
Systematically count all triangles in a composite
A big equilateral triangle (3 matchsticks per side) is split into a grid of small unit triangles. I must count every equilateral triangle of every size hidden in the figure.
Geometry
Your answer
Watch out It's easy to count only the upward-pointing triangles and forget the downward-pointing (inverted) ones tucked between them.
How to solve
Strategy Identify Subproblems — A composite figure is easiest to count by breaking it into subproblems: count by size, and within each size separate upward from downward triangles. Listing each group systematically and watching the size pattern keeps the count complete and organized.
1STEP 1
Count size-1 upward triangles
Reading the rows top to bottom, the smallest upward triangles number 1 + 2 + 3 = 6.
1 + 2 + 3 = 6
Grade 4 students can identify the unit equilateral triangles and add row by row.
4.G.A.2Make A Systematic List2STEP 2
Count size-1 downward triangles
Downward-pointing ones sit between the upward triangles: 1 in row 2, 2 in row 3, giving 1 + 2 = 3.
1 + 2 = 3
Separating downward triangles as their own subproblem makes them easy to spot and not forget.
4.G.A.2Identify Subproblems3STEP 3
Count size-2 upward triangles
A size-2 upward triangle covers 4 unit triangles; the grid fits 3 of them (no size-2 downward fits).
size-2 up = 3, size-2 down = 0
Checking each size as a separate subproblem catches the larger triangles the eye tends to skip.
4.G.A.2Identify Subproblems4STEP 4
Count the size-3 triangle and total everything
The whole figure is 1 size-3 triangle. Total: 6 + 3 + 3 + 1 = 13.
6 + 3 + 3 + 1 = 13
Organizing by size shows a clear pattern and lets the subtotals simply add to the answer.
4.G.A.2Look For A PatternThe total number of equilateral triangles in the figure equals the sum of the counts from the separate size-and-orientation groups.
Why?
The groups together hold every equilateral triangle in the figure, and no triangle sits inside two of them at once.
Why?
Each triangle has one definite size and points one definite way, so listing it under its own size and direction records it exactly once, with nothing left over.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Once the groups have their separate counts of 6, 3, 3, and 1, finding how many there are altogether is just adding those four counts.
Why?
Merging four separate piles into one total lets you push any two of them together first without changing the final sum.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Answer
13 equilateral triangles
6 + 3 + 3 + 1 = 13
13 is more than the 6 obvious smallest triangles, which is expected once you include the inverted ones, the medium size-2 triangles, and the whole outer triangle. The count is finite and well above 6 but far below the dozens you'd see in a larger grid, so it fits a side-3 figure.
Takeaway
This only needs Grade 4 shape sense and tidy counting by size you already know!
- Count size-1 upward triangles
- Count size-1 downward triangles
- Count size-2 upward triangles
- Count the size-3 triangle and total everything
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