Question

A parallelogram has sides A, B, C, and D. Sides A and B have a length of `3` and sides C and D have a length of `8`. If the angle between sides A and C is `(5 pi)/8`, what is the area of the parallelogram?

Answer 1

The area is `12×\sqrt(2+\sqrt(2))` square units, about `22.17`.

Explanation:

The area of a parallelogram equals the product of the two nonparallel sides times the some of the angle at the vertex where the two sides meet. Since all vertex angles are either congruent or supplementary to each other the sine value is always the same.

The sine of `(5\pi)/8` is found using trigonometric identities:

`\sin((5\pi)/8)=cos(\pi/8)`
`=\cos((1/2)(\pi/4))`
`=\sqrt((1+\cos(\pi/4))/2)`
`=\sqrt((1+sqrt(2)/2)/2)`
`=(\sqrt(2+\sqrt(2)))/2`

Multiply this by the product of the sides `3×8=24` to get the answer.