Systematically count all triangles in a composite

4.G.A.2 · take · grade 4

Archetype: Systematically Count Shapes in a Figure · step in a 5-type progression

Using $18$ matchsticks of equal length, the shape shown on the right was built. How many equilateral triangles of all sizes can be found in this shape in total?

Show solution

Understand

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.

Givens

A large equilateral triangle with side length 3 small triangles
It is filled with small unit equilateral triangles, both upward- and downward-pointing
18 matchsticks were used to build it

Unknowns

The total number of equilateral triangles of all sizes in the figure

Constraints

Triangles must follow the matchstick grid lines
Count both upward-pointing and downward-pointing triangles, and all sizes (side 1, 2, and 3)

Plan

#7 Identify Subproblems · also uses: #2 Make a Systematic List#5 Look for a Pattern

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.

Execute

#2 Make a Systematic List 4.G.A.2

The smallest upward-pointing triangles fill the rows. Reading top to bottom there are 1 + 2 + 3 = 6 of them.

1 + 2 + 3 = 6

Grade 4 students can identify the unit equilateral triangles and add row by row.

#7 Identify Subproblems 4.G.A.2

Between the upward unit triangles sit the inverted (downward-pointing) ones. There are 1 in the second row and 2 in the third row, giving 1 + 2 = 3.

1 + 2 = 3

Separating downward triangles as their own subproblem makes them easy to spot and not forget.

#7 Identify Subproblems 4.G.A.2

A side-2 upward triangle covers four unit triangles. The grid fits 3 of these (one in each lower-left, lower-right, and top region). There are no size-2 downward triangles because the figure is not tall enough to hold an inverted side-2 triangle.

\text{size-2 up} = 3,\quad \text{size-2 down} = 0

Checking each size as a separate subproblem catches the larger triangles the eye tends to skip.

#5 Look for a Pattern 4.G.A.2

The whole figure itself is one size-3 upward equilateral triangle. Adding all groups: 6 (size-1 up) + 3 (size-1 down) + 3 (size-2 up) + 1 (size-3 up) = 13.

6 + 3 + 3 + 1 = 13

Organizing by size shows a clear pattern and lets the subtotals simply add to the answer.

Answer: 13 equilateral triangles

Review

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.

Solve an easier related problem (tool 9): count triangles in a side-1 grid (1) and a side-2 grid (5), notice the jump, and extend the pattern to predict and check the side-3 total of 13.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Identifying and classifying the equilateral triangles of each size and orientation in the grid.

💡 This only needs Grade 4 shape sense and tidy counting by size you already know!