Question

Two opposite sides of a parallelogram have lengths of `8`. If one corner of the parallelogram has an angle of `pi/8` and the parallelogram's area is `12`, how long are the other two sides?

Answer 1

Nasty answer. Here's a walkthrough.

Explanation:

Let h be the (perpendicular) height extending from one of the bases that measures length b = 8 to the other base having the same length.

since A = bh = 8h is the area of the parallelogram, we have

8h = 12
`h = 12/8 = 3/2` is its height.

Let c be the length of one of the unknown sides.
From basic trigonometry,

`sin(pi/8) = h/c`.

We may obtain the value of `sin(pi/8)` from the half-angle formula for sine.

That value is
`sin(pi/8) = sqrt(2 -sqrt(2))/2`

So that...

`sqrt(2 -sqrt(2))/2 = (3/2)/c`

Cross multiply:

`csqrt(2 -sqrt(2)) = 3`

`c = 3/(sqrt(2 -sqrt(2)))`

If we demand that the expression be rationalized, then
multiply first by
`sqrt(2 -sqrt(2))/sqrt(2 -sqrt(2))` and next by
`(2 + sqrt(2))/(2 +sqrt(2))`.

Answer 2

Picture

Explanation:

If you also have the formula

`A = bcsin(alpha)`

where `alpha` is the angle included between sides b and c, you may get there more quickly.

`12 = 8csin(pi/8)`.

Use the value of the sine, and it comes out quickly.