Question

Find the number of triangles in the given figure. 

 

• A. 18
• B. 20
• C. 24
• D. 27
• E. 30

Hint:

When $n$ is even, use $N = \frac{n(n+2)(2n+1)}{8}$. When $n$ is odd, use $N = \frac{n(n+2)(2n+1)-1}{8}$. This formula is the fastest way to solve "triangle-in-triangle" grid problems.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To count triangles in a large equilateral triangle divided into smaller ones (a triangular grid), we use a systematic approach based on the number of "base units" ($n$). We must count triangles of all sizes: 1x1, 2x2, 3x3, etc.
Step 2: Key Formula or Approach:
For a triangle with $n$ divisions on the side, the formula for the total number of triangles is:
\[ N = \frac{n(n+2)(2n+1)}{8} \]
Note: If the result is not an integer, take the floor value (round down).
Step 3: Detailed Explanation:
Assuming the figure is a large triangle with 4 smaller units on each side ($n=4$):
1. Triangles of size 1 (Smallest): 16
2. Triangles of size 2: 7 (upward) + 0 (downward) = 7
3. Triangles of size 3: 3
4. Triangles of size 4 (Largest): 1
Using the formula for \(n=4\): \[ N = \frac{4(4+2)(2 \times 4 + 1)}{8} = \frac{4 \times 6 \times 9}{8} = \frac{216}{8} = 27 \]
Step 4: Final Answer:
The total number of triangles in the figure is 27.