I just ran across this in an old Mathematical Gazette: R.H. Macmillan of Buckinghamshire shared a tidy expression for the area of a triangle whose vertices have coordinates (x1, y1), (x2, y2), and (x3, y3):

\displaystyle  \pm \frac{1}{2}\left \{ x_{1} \left ( y_{2} - y_{3} \right ) + x_{2} \left ( y_{3} - y_{1} \right ) + x_{3} \left ( y_{1} - y_{2} \right ) \right \}

The sign is positive if the numbering is counterclockwise and negative if it’s clockwise.

“The expression is readily derived geometrically (using only the fact that the sum of the areas on each side of the diagonal of a rectangle must be equal) and so provides an interesting elementary exercise.”

(R.H. Macmillan, “Area of a Triangle,” Mathematical Gazette 77:478 (March 1993), 88.)