Signed Area
<p>Rectangle: area = base × height
Triangle: area = base × height / 2
Trapazoid: area = base × (height_1 + height_2) / 2
Circle: "area = Pi × r² = Pi × diameter² / 4" (This is going to be replaced.)
Triangle (Heron or Archimedes) surface ≡ (a + b + c)/2 (also called a semi-perimeter) whose area is 'area = √[(s(s - a)(s - b)(s - c)]' (This is going to be replaced.)
None of these formulas are used in Affine Algebraic Geometry.
The standard treatment rests on the rectangle area formula: area = base × height.
Why are these formulas not being used.
1) Consistency is not present (areas are not additive and are not rotationally invariant).
2) Consistency of areas when they may overlap, cross, or cancel.
3) Not flexible for complex geometrical combination (complex geometry does not subdivide into rectangles [or triangles out of which they are derived]).
 
Affine Geometry is the key:
 
Signed area in terms of cross products:
 
s(OAB) = 1/2 (ad - bc)
            = 1/2 u × v
            = 1/2 OA × OB
 
more generally of an arbitrary oriented triangle:
 
s(ABC) = 1/2 AB × AC
            = 1/2 BC × BA
            = 1/2 CA × CB</p>