Area of Trapezoid

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Theorem

TrapezoidArea.png

Let $ABCD$ be a trapezoid:

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 $BECF$ 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