Question

How do you find the area of a parallelogram?

Answer 1

The two easiest ways of finding the area of a paralelogram with side lenghts `a` and `b` and heights `h` and `h'` are:

`"Area" = {(ab sinA=ab sinC=ab sinB=ab sinD),(ah=bh') :}`

Explanation:

There are many ways of expressing the area of a parallelogram. Here's a few.

Let us have a parallelogram `ABCD` and let `S` be its area

We can write `S` as:

`S = S_(DeltaABC)+S_(DeltaADC)=2S_(DeltaABC)`

One formula to calculate the area of a triangle is

`"Area" = 1/2ab sinC`

`S = 2*1/2AB*BCsinC`

`color(red)(S = AB*BCsinA=AB*BCsinC`

We can also say `S` is the sum of the areas of the triangles `Delta ABD` and `DeltaBCD` to prove that

`color(red)(S = AB*BCsinB=AB*BCsinD`

Another way, without the use of trigonometric identities, is in terms of the heights.

Let `h` be the height coming down from `A` to the side `CD`, and let `a`, `b` be the lenghts of the sides of the parallelogram. Then, we can write the area of `ABCD` as the area of two rectangles:

`S=(a-x)h+xh=ah`

Analogously, if `h'` is the other height, then

`S=bh'`

Answer 2

The diagonal of a parallelogram divides it into two congruent triangles, so all the formulas for the area of a triangle apply.