Question

Two opposite sides of a parallelogram each have a length of `6`. If one corner of the parallelogram has an angle of `( pi)/3` and the parallelogram's area is `24`, how long are the other two sides?

Answer 1

The lengths of the other two sides are both `8/sqrt3`.

Explanation:

Call the parallelogram `ABCD` and let `AB=CD=6` and `m/_A=m/_C=pi/3`. We know the area of the parallelogram can be calculated by the following formula using vectors: `A=|vec (AB) xx vec (AD)|=|vec (AB)||vec (AD)|sintheta`. We are given that the area is 24, so
`24=|vec (AB)||vec (AD)|sintheta`
`24=6|vec (AD)|sin(pi/3)`
`|vec (AD)|=4/(sqrt3/2)`
`:.|vec (AD)|=8/sqrt3`
Since the lengths of opposite sides in a parallelogram are equal, we know that the lengths of the other two sides are `8/sqrt3`.