Area of Parallelogram/Rectangle

Theorem
The area of a rectangle equals the product of one of its bases and the associated altitude.

Proof
Let $ABCD$ be a rectangle.


 * Area-of-Rectangle.png

Then construct the square with side length:
 * $\map \Area {AB + BI}$

where $BI = BC$, as shown in the figure above.

Note that $\square CDEF$ and $\square BCHI$ are squares.

Thus:
 * $\square ABCD \cong \square CHGF$

Since congruent shapes have the same area:
 * $\map \Area {ABCD} = \map \Area {CHGF}$ (where $\map \Area {FXYZ}$ denotes the area of the plane figure $FXYZ$).

Let $AB = a$ and $BI = b$.

Then the area of the square $AIGE$ is equal to: