Step 1: Assume dimensions using the given ratio

Since the sides of the rectangle are in the ratio $8: 10$, let the length be $10x$ and the breadth be $8x$.

$\text{Length } l = 10x, \quad \text{Breadth } b = 8x$

Step 2: Use the perimeter formula to find $x$

The formula for the perimeter of a rectangle is $\text{Perimeter} = 2(l + b)$. Substituting the given perimeter of $90\text{ cm}$:

$2(10x + 8x) = 90 \implies 36x = 90 \implies x = \frac{90}{36} = 2.5$

Step 3: Calculate the actual dimensions

Substitute $x = 2.5$ back into the expressions for length and breadth:

$l = 10 \times 2.5 = 25\text{ cm}, \quad b = 8 \times 2.5 = 20\text{ cm}$

Step 4: Calculate the area of the rectangle

Using the area formula $\text{Area} = \text{length} \times \text{breadth}$:

$\text{Area} = 25 \times 20 = 500\text{ cm}^2$