# Area of Trapezoid

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## Theorem

Let $ABCD$ be a trapezoid:

and

- whose height is $h$.

Then the area of $ABCD$ is given by:

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

## Proof

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

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

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

Also, $AEFD$ is a parallelogram which comprises the two trapezoids $ABCD$ and $BECF$.

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

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

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 4$: Geometric Formulas: $4.7$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes