Question

How do you derive the formula for a triangle?

Answer 1

Probably, the question is about a formula for an area of a triangle.
If not, ask more specifically.
Below is the proof of a formula `S=1/2*a*h`,
where `a` is a base and `h` is an altitude.

Explanation:

Consider triangle `Delta ABC` below

An altitude `h` separates our triangle into two right triangles.
As seen from the picture, the area of each one of these right triangles is doubled, so the area of a rectangle with the same base `a` and altitude `h` is twice the area of an initial triangle.

Since the area of a rectangle is `a*h`, the area of the initial triangle is
`S = 1/2*a*h`

The drawing will be different in case of a triangle with an obtuse angle at the base. In this case an analogous transformation into a rectangle will represent our triangle not as a sum of two right triangles, but as their difference with the same final formula.