How many triangles are there in the given figure?

Concept

A triangle is any closed three-sided region whose three sides each run entirely along lines that are actually drawn. To count every triangle without missing or repeating any, work by SIZE: first count the smallest atomic triangles, then count the larger triangles that are built by joining two or more adjacent smaller ones along a shared line.

Applying it to this figure

The figure is a central square split into a 2×2 grid (one horizontal and one vertical midline) with a diamond inscribed by joining the four side-midpoints, plus one triangular wing attached on the left and one on the right. Count level by level:

1. Smallest triangles: inside the square, each of the 4 grid cells is cut by one diamond edge into 2 triangles, giving 4 × 2 = 8 smallest triangles.

2. Half-diamond triangles: joining two adjacent smallest triangles that share a diamond diagonal gives the four half-diamond triangles — top half, bottom half, left half and right half of the diamond. That is 4 triangles.

3. Wing triangles: the left wing is one triangle and the right wing is one triangle, giving 2 more.

Total = 8 + 4 + 2 = 14 triangles.

Cross-check

No larger composite triangle is created where a wing meets the square: a wing's slanted edge and the square's side do not continue into one straight line, so they cannot serve as a single side of a bigger triangle. Inside the square the only slanted lines are the four diamond edges, and the square has no corner-to-corner diagonals, so there are no half-square triangles. Re-counting by size confirms 8 + 4 + 2 with nothing left over, so the figure contains exactly 14 triangles.