Perimeter of Trapezoid

Theorem

 * TrapezoidPerimeter.png

Let $ABCD$ be a trapezoid:
 * whose parallel sides are of lengths $a$ and $b$
 * whose height is $h$.

and
 * whose non-parallel sides are at angles $\theta$ and $\phi$ with the parallels.

The perimeter $P$ of $ABCD$ is given by:
 * $P = a + b + h \left({\csc \theta + \csc \phi}\right)$

where $\csc$ denotes cosecant.

Proof
The perimeter $P$ of $ABCD$ is given by:
 * $P = AB + BC + CD + AD$

where the lines are used to indicate their length.

Thus:

Hence:

Hence the result.