Given:
Perimeter of the rhombus = 64 m
Length of one diagonal = 22 m

To Prove:
Area of the rhombus = 66√15 sq. meters

Proof:

Step 1: Finding the Side Length of the Rhombus
Let's assume the side length of the rhombus as 'a'. Since a rhombus has all sides equal, the perimeter can be expressed as:
Perimeter = 4 * a

Given that the perimeter is 64 m, we can write the equation:
4 * a = 64

Simplifying the equation, we get:
a = 64 / 4 = 16 m

So, the side length of the rhombus is 16 m.

Step 2: Finding the Other Diagonal of the Rhombus
Let's assume the other diagonal of the rhombus as 'd'. Using the properties of a rhombus, we know that the diagonals bisect each other at right angles. Therefore, each half of the diagonal forms a right-angled triangle with the side lengths 'a' and 'd/2'.

Using the Pythagorean theorem, we can write the equation:
(a/2)^2 + (d/2)^2 = (22/2)^2

Substituting the value of 'a' as 16 m, we get:
(16/2)^2 + (d/2)^2 = 11^2
8^2 + (d/2)^2 = 121
64 + (d/2)^2 = 121
(d/2)^2 = 121 - 64
(d/2)^2 = 57

Taking the square root of both sides, we get:
d/2 = √57

Multiplying both sides by 2, we get:
d = 2 * √57

So, the length of the other diagonal is 2√57 m.

Step 3: Calculating the Area of the Rhombus
The area of a rhombus can be calculated using the formula:
Area = (d1 * d2) / 2
where d1 and d2 are the lengths of the diagonals.

Substituting the values of the diagonals, we get:
Area = (22 * 2√57) / 2
Area = 22√57

Calculating the numerical value of the area, we have:
Area ≈ 66√15 sq. meters

Therefore, the area of the rhombus is approximately 66√15 sq. meters.