Area of Trapezium

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Theorem

TrapezoidArea.png

Let $ABCD$ be a trapezium:

whose parallel sides are of lengths $a$ and $b$

and

whose height is $h$.


Then the area of $ABCD$ is given by:

$\Box ABCD = \dfrac {h \paren {a + b} } 2$


Proof

TrapezoidAreaProof.png

Extend line $AB$ to $E$ by length $a$.

Extend line $DC$ to $F$ by length $b$.

Then $BEFC$ is another trapezium whose parallel sides are of lengths $a$ and $b$ and whose height is $h$.

Also, $AEFD$ is a parallelogram which comprises the two trapezia $ABCD$ and $BEFC$.

So $\Box ABCD + \Box BEFC = \Box AEFD$ and $\Box ABCD = \Box BEFC$.

$AEFD$ is of altitude $h$ with sides of length $a + b$.

Thus from Area of Parallelogram the area of $AEFD$ is given by:

$\Box AEFD = h \paren {a + b}$

It follows that $\Box ABCD = \dfrac {h \paren {a + b} } 2$

$\blacksquare$


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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