Parallelogram Area

Parallelogram Area

Say we want to find the area of a parallelogram with base $b$ and altitude $h:$

We can simply rearrange the parts of the parallelogram:

What shape is this?

It's a rectangle! We know how to find this area; it's just $\text{base} \cdot \text{height} = bh.$ So the area of a parallelogram with base of length $b$ and height of length $h$ is given by the formula $\text{Area} = b\cdot h$.

There is a parallelogram below. Sides $CB$ and $AD$ are parallel, and so are sides $CA$ and $BD$. $\angle{ACF}$ and $\angle{BDE}$ are equal, and all the right angles depicted are right. So let's call the length of side $CA=BD=X$, the length of $BE=ED=Y$, the length of $CF=ED=Z$, and the length of $FB=AE=W$. Now notice that $\bigtriangleup FCA$ has the same area as $\bigtriangleup BED$, and if we attach them to make a rectangle, the rectangle has area $YZ$. Obviously, the area of rectangle $AFBE$ is $WY$, so if we add the areas of these two rectangles, we get $Y(W + Z)$.

Therefore that is our area for our parallelogram $ABCD$, and we have derived a formula for An Area Of A Parallelogram: $Y(W + Z)$. In other words, the area of a $ABCD$, $(ABCD)=bh$ where $b$ is the base and $h$ is the height considering $b$ is the base.