Triangles

This is a simple counting exercise; just make sure you don't miss any!

triangles
Solution

The total number triangles is 27.

The triangles split into two groups nicely; those with a point at the top (like the red triangles) and those with a point at the bottom (like the white triangles)
Point at the top
There is 1 triangle with side length 4
There are 3 triangles with side length 3
There are 6 triangles with side length 2
There are 10 triangles with side length 1
Point at the bottom
There is 1 triangle with side length 2
There are 6 triangles with side length 1

The total number is therefore $$T = (1 + 3 + 6 + 10) + (1 + 6 ) = 27$$