Perimeter of Trapezoid

Theorem

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:

$P = a + b + h \paren {\csc \theta + \csc \phi}$

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:

\(\text {(1)}: \quad\)

\(\ds AB\)

\(=\)

\(\ds b\)

\(\text {(2)}: \quad\)

\(\ds CD\)

\(=\)

\(\ds a\)

\(\ds h\)

\(=\)

\(\ds AD \sin \theta\)

Definition of Sine of Angle

\(\ds \leadsto \ \ \)

\(\ds AD\)

\(=\)

\(\ds \frac h {\sin \theta}\)

\(\text {(3)}: \quad\)

\(\ds \)

\(=\)

\(\ds h \csc \theta\)

Cosecant is Reciprocal of Sine

\(\ds h\)

\(=\)

\(\ds BC \sin \phi\)

Definition of Sine of Angle

\(\ds \leadsto \ \ \)

\(\ds BC\)

\(=\)

\(\ds \frac h {\sin \phi}\)

\(\text {(4)}: \quad\)

\(\ds \)

\(=\)

\(\ds h \csc \phi\)

Cosecant is Reciprocal of Sine

Hence:

\(\ds P\)

\(=\)

\(\ds AB + BC + CD + AD\)

\(\ds \)

\(=\)

\(\ds b + h \csc \phi + a + h \csc \theta\)

from $(1)$, $(2)$, $(3)$ and $(4)$

Hence the result.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: $4.8$