The base of the pyramid is a parallelogram. Diagonals of its 8 and 4 root of 3 angle between diagonals 30 is the height of the pyramid

The base of the pyramid is a parallelogram. Diagonals of its 8 and 4 root of 3 angle between diagonals 30 is the height of the pyramid equal to the larger side of the parallelogram volume?

Given:
Pyramid;
The base is a parallelogram;
Diagonals – 8 and 4 √3;
Angle α = 30 degrees;
Height = large side of the base of AD;
Find volume:
Decision:
t. О is the point of intersection of the diagonals, where ОА = ОС = 8/2 = 4, ОВ = ОD= 4 √3 / 2 = 2 √3;
Consider the triangle AOD by the cosine theorem:
Angle β = Angle AOD = 180 degrees – 30 degrees. = 150 degrees;
Then, we find the volume:
V = 1/3 * S main * h, where S main = 1/2 * AC * BD * sin α;
Substituting the known data, we get:
S = 1/2 * 8 * 4 * √3 * 1/2 = 4 * 2 √3 = 8 √3;
Hence, by the cosine theorem, we obtain:
a ^ 2 = 4 ^ 2 + (2 √3) ^ 2 – 2 * 4 * 2 √3 * cosβ;
a ^ 2 = 16 + 4 * 3 – 16 * √3 (- √3 / 2) = 16 + 12 + 3 * 8 = 28 + 24 = 52;
a = √52;
Since h = a, then:
V = 1/3 * 8 √3 * √52 = 8/3 * √156 = 33. 3;
Answer: approximately 33.3