Hexagons in 6-Row Equilateral Triangle: 11 Total

Answer

In a large equilateral triangle made of small unit triangles with $n$ rows, a regular hexagon of side length $k$ can only exist if $n \geq 3k$. For a triangle with $n=6$ rows, there are 10 hexagons of side length $s=1$ and 1 hexagon of side length $s=2$, totaling 11 hexagons.

Explanation

I see a series of diagrams showing a large equilateral triangle divided into smaller unit triangles. Fig. 2 shows a 5-row triangle where small hexagons (side 1) are highlighted. Fig. 3 shows a 6-row triangle containing a larger hexagon with a side length of 2 units.

1. Understanding Hexagon Side Lengths A regular hexagon is composed of 6 small equilateral triangles meeting at a center point. To form a hexagon of side length $s$, the height of the hexagon (from top vertex to bottom vertex) requires $3s$ rows of the small triangles. $n \geq 3s$ This inequality shows that the number of rows $n$ must be at least three times the side length $s$ of the hexagon we want to find.

2. Counting Hexagons of Side Length 1 A hexagon of side length $s=1$ requires at least 3 rows to exist. In a triangle of $n$ rows, the number of possible center points for these hexagons forms a smaller triangle. The number of hexagons of side $s=1$ is the $(n-2)$-th triangular number. $T_{n-2} = \frac{(n-2)(n-1)}{2}$ For $n=6$, we calculate $T_{6-2} = T_4 = 1+2+3+4 = 10$. This formula calculates the sum of integers from 1 up to $n-2$ to find the total count.

3. Counting Hexagons of Side Length 2 A hexagon of side length $s=2$ requires at least $3 \times 2 = 6$ rows. If $n=6$, there is only enough room for exactly one such hexagon. The formula for hexagons of side length $s$ in a triangle of $n$ rows is: $H_s = \frac{(n-3s+1)(n-3s+2)}{2}$ For $n=6$ and $s=2$, we get $H_2 = \frac{(6-6+1)(6-6+2)}{2} = \frac{1 \times 2}{2} = 1$. This formula represents the number of valid positions a hexagon of size $s$ can occupy within the larger grid.

4. Summing the Total To find the total number of hexagons in a triangle with 6 rows, we add the counts of all possible side lengths ($s=1$ and $s=2$). $\text{Total} = 10 + 1 = 11$ The total is the sum of all hexagons of various sizes that fit within the given boundary.

Final Answer

In a triangle with six rows, the total number of regular hexagons is: $\boxed{11}$

Common Mistakes

• Forgetting the height requirement: Students often think a side-2 hexagon can fit in a 4 or 5-row triangle, but it actually requires 6 full rows of vertical space to complete the shape.
• Incorrect Triangular Numbers: When counting, students might accidentally use $n$ or $n-1$ instead of $n-2$ for the base of the sum, leading to over-counting.