Originally Posted by

**earboth** If you have the points $\displaystyle P(x_P, y_P)$ and $\displaystyle Q(x_Q, y_Q)$ then the distance $\displaystyle |\overline{PQ}| = \sqrt{(x_P - x_Q)^2+(y_P-y_Q)^2}$.

One of the many applications of the Pythagorean theorem.

If the vertices of a polygon are determined by their coordinates then you can calculate the area by the chop-off method: You choose a rectangle whose sides are parallel to the axes and who contains completely the area in question (see attachment). Then you "cut off" those right triangles (or trapezods) which don't belong to the area. What's left is the value of the area.

With my example:

The surrounding rectangle has the area $\displaystyle a_r = 5 \cdot 6 = 30$

The areas of the triangles $\displaystyle \Delta_1 = 7.5, \Delta_2 = 4.5, \Delta_3 = 6$

Subtract the areas of the triangles from the area of the rectangle : 30 -18 = 12.

So the red-striped triangle has an area of 12 (units)