Area of Parallelogram/Parallelogram

Theorem
Let $ABCD$ be a parallelogram whose adjacent sides are of length $a$ and $b$ enclosing an angle $\theta$.

The area of $ABCD$ equals the product of one of its bases and the associated altitude:

where:
 * $b$ is the side of $ABCD$ which has been chosen to be the base
 * $h$ is the altitude of $ABCD$ from $b$.

Proof

 * Area-of-Parallelogram.png

Let $ABCD$ be the parallelogram whose area is being sought.

Let $F$ be the foot of the altitude from $C$

Also construct the point $E$ such that $DE$ is the altitude from $D$ (see figure above).

Extend $AB$ to $F$.

Then:

Thus:
 * $\triangle AED \cong \triangle BFC \implies \map \Area {AED} = \map \Area {BFC}$

So: