← Count composite triangles by size and orientation · Systematically Count Shapes in a Figure

Count composite triangles by size and orientation · 4 practice problems

4.G.A.2

Generated variants — 4

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 5 equilateral triangles

Using $9$ 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

1 · Understandwhat's really being asked

A big equilateral triangle (2 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 2 small triangles
It is filled with small unit equilateral triangles, both upward- and downward-pointing
9 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

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Count the smallest (size-1) upward triangles

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

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

1 + 2 = 3

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

2Count the smallest (size-1) downward triangles

#7 Identify Subproblems 4.G.A.2

Between the upward unit triangles sit the inverted (downward-pointing) ones. Counting row by row gives 1.

1 = 1

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

3Count every larger size

#7 Identify Subproblems 4.G.A.2

A size-k triangle covers k*k unit triangles. Checking each larger size as its own subproblem, the upward counts by size are 3 + 1 and the downward counts by size are 1.

\text{up}: 3 + 1 = 4;\quad \text{down}: 1 = 1

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

4Add every group for the total

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

Add all upward triangles (4) and all downward triangles (1) of every size.

4 + 1 = 5

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

Answer: 5 equilateral triangles

4 · Reviewdoes it hold up?

5 is more than the 3 obvious smallest upward triangles, which is expected once you include the inverted ones and every larger size up to the whole outer triangle. The count is finite and grows quickly with the side length.

Another way: Solve easier related problems (tool 9): count triangles in a side-1 grid (1) and a side-2 grid (5), notice the jump, and extend the pattern up to the side-2 total of 5.

Standardsmin 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.

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

Variant 2 medium answer: 13 equilateral triangles

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

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

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

3 · Execute4 carry out the plan

1Count the smallest (size-1) upward triangles

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

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

1 + 2 + 3 = 6

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

2Count the smallest (size-1) downward triangles

#7 Identify Subproblems 4.G.A.2

Between the upward unit triangles sit the inverted (downward-pointing) ones. Counting row by row gives 1 + 2.

1 + 2 = 3

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

3Count every larger size

#7 Identify Subproblems 4.G.A.2

A size-k triangle covers k*k unit triangles. Checking each larger size as its own subproblem, the upward counts by size are 6 + 3 + 1 and the downward counts by size are 3.

\text{up}: 6 + 3 + 1 = 10;\quad \text{down}: 3 = 3

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

4Add every group for the total

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

Add all upward triangles (10) and all downward triangles (3) of every size.

10 + 3 = 13

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

Answer: 13 equilateral triangles

4 · Reviewdoes it hold up?

13 is more than the 6 obvious smallest upward triangles, which is expected once you include the inverted ones and every larger size up to the whole outer triangle. The count is finite and grows quickly with the side length.

Another way: Solve easier related problems (tool 9): count triangles in a side-1 grid (1) and a side-2 grid (5), notice the jump, and extend the pattern up to the side-3 total of 13.

Standardsmin 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.

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

Variant 3 medium answer: 27 equilateral triangles

Using $30$ 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

1 · Understandwhat's really being asked

A big equilateral triangle (4 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 4 small triangles
It is filled with small unit equilateral triangles, both upward- and downward-pointing
30 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

2 · Planchoose the strategy

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

3 · Execute4 carry out the plan

1Count the smallest (size-1) upward triangles

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

The smallest upward-pointing unit triangles fill the rows. Reading top to bottom there are 1 + 2 + 3 + 4 of them.

1 + 2 + 3 + 4 = 10

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

2Count the smallest (size-1) downward triangles

#7 Identify Subproblems 4.G.A.2

Between the upward unit triangles sit the inverted (downward-pointing) ones. Counting row by row gives 1 + 2 + 3.

1 + 2 + 3 = 6

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

3Count every larger size

#7 Identify Subproblems 4.G.A.2

A size-k triangle covers k*k unit triangles. Checking each larger size as its own subproblem, the upward counts by size are 10 + 6 + 3 + 1 and the downward counts by size are 6 + 1.

\text{up}: 10 + 6 + 3 + 1 = 20;\quad \text{down}: 6 + 1 = 7

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

4Add every group for the total

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

Add all upward triangles (20) and all downward triangles (7) of every size.

20 + 7 = 27

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

Answer: 27 equilateral triangles

4 · Reviewdoes it hold up?

27 is more than the 10 obvious smallest upward triangles, which is expected once you include the inverted ones and every larger size up to the whole outer triangle. The count is finite and grows quickly with the side length.

Another way: Solve easier related problems (tool 9): count triangles in a side-1 grid (1) and a side-2 grid (5), notice the jump, and extend the pattern up to the side-4 total of 27.

Standardsmin 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.

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

Variant 4 hard answer: 48 equilateral triangles

Using $45$ 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

1 · Understandwhat's really being asked

A big equilateral triangle (5 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 5 small triangles
It is filled with small unit equilateral triangles, both upward- and downward-pointing
45 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

2 · Planchoose the strategy

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

3 · Execute4 carry out the plan

1Count the smallest (size-1) upward triangles

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

The smallest upward-pointing unit triangles fill the rows. Reading top to bottom there are 1 + 2 + 3 + 4 + 5 of them.

1 + 2 + 3 + 4 + 5 = 15

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

2Count the smallest (size-1) downward triangles

#7 Identify Subproblems 4.G.A.2

Between the upward unit triangles sit the inverted (downward-pointing) ones. Counting row by row gives 1 + 2 + 3 + 4.

1 + 2 + 3 + 4 = 10

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

3Count every larger size

#7 Identify Subproblems 4.G.A.2

A size-k triangle covers k*k unit triangles. Checking each larger size as its own subproblem, the upward counts by size are 15 + 10 + 6 + 3 + 1 and the downward counts by size are 10 + 3.

\text{up}: 15 + 10 + 6 + 3 + 1 = 35;\quad \text{down}: 10 + 3 = 13

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

4Add every group for the total

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

Add all upward triangles (35) and all downward triangles (13) of every size.

35 + 13 = 48

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

Answer: 48 equilateral triangles

4 · Reviewdoes it hold up?

48 is more than the 15 obvious smallest upward triangles, which is expected once you include the inverted ones and every larger size up to the whole outer triangle. The count is finite and grows quickly with the side length.

Another way: Solve easier related problems (tool 9): count triangles in a side-1 grid (1) and a side-2 grid (5), notice the jump, and extend the pattern up to the side-5 total of 48.

Standardsmin 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.

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