# 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.

Then construct the square with side length:

$\paren {AB + BI}$

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:

$\paren {ABCD} = \paren {CHGF}$ (where $\paren {FXYZ}$ denotes the area of the plane figure $FXYZ$).

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

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

 $\displaystyle \paren {a + b}^2$ $=$ $\displaystyle a^2 + 2 \paren {ABCD} + b^2$ $\displaystyle \paren {a^2 + 2 a b + b^2}$ $=$ $\displaystyle a^2 + 2 \paren {ABCD} + b^2$ $\displaystyle a b$ $=$ $\displaystyle \paren {ABCD}$

$\blacksquare$