The perimeter of the rhombus is 24, and the cosine of one of the corners is 2 √2 and everything

The perimeter of the rhombus is 24, and the cosine of one of the corners is 2 √2 and everything divided by 3. Find the area of the rhombus.

If the perimeter of a rhombus is 24, then its side is:

a = P / 4 = 24/4 = 6.

Knowing the cosine of the angle and using the identity sin2α + cos2α = 1, we find the sine of the rhombus angle:

sin2α = 1 – cos2α = 1 – (2√2 / 3) ^ 2 = 1 – 4 * 2/9 = 1 – 8/9 = 1/9;

sinα = √ (1/9) = 1/3.

The area of a rhombus is equal to the product of the square of the side and the sine of the angle between them:

S = a ^ 2 * sinα = 6 ^ 2 * 1/3 = 36/3 = 12.