Question

Two opposite sides of a parallelogram each have a length of `8`. If one corner of the parallelogram has an angle of `( pi)/3` and the parallelogram's area is `80`, how long are the other two sides?

Answer 1

Length `b=10sqrt2`.

Explanation:

Solving for:
Length `b`

Given:
Area `A=80`
Length `a=8`
Angle `angleC=pi/3`

The area of a parallelogram is equal to the product `atimesh`, where `h` is the height (distance) between the `a`-sides.

`"        "A=ah`
`=>80=8h`
`=>10=h "      "therefore` the height is 10.

The side we're solving for, `b`, can be considered the hypotenuse of a right triangle with one angle `pi/3` and its opposite side with length `10`.

`"      "sin" "angle" "C""="opp"/"hyp"`

`=>sin(pi/3)=10/b`

`=>"   "sqrt2/2"    "=10/b`

`=>"       "b"     "=(2*10)/sqrt2color(blue)(*sqrt2/sqrt2)`        (rationalize)

`=>"       "b"     "=(20sqrt2)/2=10sqrt2`

`therefore` the length `b` is `10sqrt2`.

Bonus:

From this, we can actually get a general formula that relates a parallelogram's area to its side lengths and the one angle:

Since `sinangleC=h/b`, we can solve this for `h` to get
`h=bsinangleC`

And since we know `A=ah`, we can plug the above RHS in for `h`:

`"       "A=ah`
`=>A=ab sinangleC`

You may have seen a similar formula for the area of a triangle:
`A_triangle=1/2ab sinC`
And we know a parallelogram can be geometrically split into two identical triangles, thus, twice the area of such a triangle is the area of the whole parallelogram.