Perimeter of Trapezium

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Theorem

TrapezoidPerimeter.png

Let $ABCD$ be a trapezium:

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.

$\blacksquare$


Usage Differences

The North American definitions of trapezium and trapezoid differ from most of the rest of the world as follows:

a trapezoid has one pair of sides that are parallel
a trapezium does not have a pair of parallel sides.

This is the opposite way round from the definitions as used in most of the rest of the world, as used by $\mathsf{Pr} \infty \mathsf{fWiki}$.


In order to reduce confusion, when a trapezoid is intended, it may be better to use the term irregular quadrilateral instead of trapezoid.


It is worth noting that Euclid, in his definitions, did not distinguish between trapezia and trapezoids, and lumped them together as trapezia:

In the words of Euclid:

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

(The Elements: Book $\text{I}$: Definition $22$)


Sources

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