The acute angle of the rhombus is 60 degrees, and its area is 54√3. Find the length of the larger diagonal of the rhombus.

We know that: S = 54√3;

Let’s use the formula S = a ^ 2 * sin a;

Let’s substitute:

54√3 = a ^ 2 * sin a;
a ^ 2 = 54√3 / sin a;
a ^ 2 = 54√3 / (√3 / 2) = 54√3 * 2 / √3 = 54 * 2 = 108;
a = √108 = 4√7;

angle AOB = 90 °, because diagonals – bisectors then angle ABO = 30 °;
angle AOB = 90 °, ABO = 30 °, item 1 side a = 4√7;

cos ABO = BO / AB;
BO = AB * cos ABOBO = 4√7 * cos 30 = 4√7 * √3 / 2 = 2 * √21;
BD = 2 * BO;
BD = 2 * 2√21 = 4√21.

Answer: 4√21.