The area of the rectangle = 36√3 cm², the angle between the diagonals is 60 °. Find sides.

The area of the parallelogram (across the diagonals) is S = 1/2 * d1 * d2 * sina (d1 and d2 are diagonals, a is an acute angle between them).

A rectangle is a parallelogram with equal diagonals.

S = 1/2 * d² * sina = 36√3.

d² * sin60 ° = 72√3.

d² * √3 / 2 = 72√3.

d² = 144.

d = 12cm.

The two sides of the rectangle and the diagonal form a right-angled triangle, one of the corners of which is 60 °. This means that one side of the triangle is 12 * sin60 ° = 12 * √3 / 2 = 6√3 (cm). And the second side is 12 * cos60 ° = 12 * 1/2 = 6 (cm).