How many triangles are in the given shape?

The correct answer is Option (3) → 27

The figure is a large triangle cut into 4 rows of small triangles. Count the triangles pointing up and those pointing down separately, by size (a side of 1, 2, 3 or 4 small triangles).

Triangles pointing up:

• Size 1: \(1+2+3+4=10\)

• Size 2: \(1+2+3=6\)

• Size 3: \(1+2=3\)

• Size 4: the whole figure, 1

• Total pointing up \(=10+6+3+1=20\)

Triangles pointing down (shown in the picture):

• Size 1: \(1+2+3=6\) (amber)

• Size 2: only 1, which needs two rows below it (green)

• Total pointing down \(=7\)

Total number of triangles \(=20+7=27\).

Shortcut: For a triangle divided into \(n\) rows with \(n\) even, the count is \(\displaystyle \frac{n(n+2)(2n+1)}{8}\). With \(n=4\): \(\displaystyle \frac{4\times6\times9}{8}=27\).