Shoelace Formula
Triangle Area (Counter-Clockwise)
Finding the Area of the Triangle
We can use the trapezoids to find the area of a triangle where all of the sides are oblique (neither vertical nor horizontal).
Consider the areas of each of the three trapezoids in the diagram above:
But we can use these to find the area of
What if we were to switch points and ?
What would happen?
Triangle Area (Clockwise)
Swapping B and C to make the orientation Clockwise
How have the trapezoid areas changed?
Consider the areas of each of the three trapezoids in the diagram above:
This means we got the same result from before, but this time it came out negative. To fix this, we can simply use absolute value.
Shoelace Formula
The area of a triangle with vertices , , and is:
These three products can be visualized with a shoelace pattern by writing out the three vertices, in order, and returning to the first vertex. All of the products of the diagonals in one direction are added, and then we subtract the products on the diagonals going the other direction. Don't forget to take the absolute value and multiply by !
Shoelace for Quadrilateral
The formula can also be applied to Quadrilaterals or other polygons in general. Try dragging A,B,C,D around to see what happens if the quadrilateral is not convex, or if it is self-intersecting. Does it seem like the formula is still valid?