Introduction:
In this problem, we are given the length of one side of a rhombus as 5 cm and one of its diagonals as 8 cm. We need to find the area of the rhombus.

Step-by-Step Solution:

1. Understand the properties of a rhombus:
A rhombus is a quadrilateral with all sides of equal length. It has two pairs of opposite congruent angles. The diagonals of a rhombus bisect each other at right angles.

2. Draw and label the given information:
Let's draw a rhombus and label the given information.

_______
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/_____________________________\

We are given:
- One side of the rhombus = 5 cm
- One diagonal of the rhombus = 8 cm

3. Use the properties of a rhombus to find the other side:
Since a rhombus has all sides of equal length, we can conclude that the length of the opposite side is also 5 cm.

4. Use the properties of a rhombus to find the lengths of the other diagonal:
Since the diagonals of a rhombus bisect each other at right angles, we can use the Pythagorean theorem to find the length of the other diagonal.

Let the length of the other diagonal be 'd'.

Using the Pythagorean theorem, we have:
(5/2)^2 + (d/2)^2 = 8^2
25/4 + (d/2)^2 = 64
(d/2)^2 = 64 - 25/4
(d/2)^2 = 256/4 - 25/4
(d/2)^2 = 231/4
d/2 = sqrt(231/4)
d/2 = sqrt(231)/2
d = 2 * sqrt(231)/2
d = sqrt(231)

Therefore, the length of the other diagonal is sqrt(231) cm.

5. Calculate the area of the rhombus:
The area of a rhombus can be calculated using the formula:
Area = (diagonal1 * diagonal2)/2

Substituting the given values, we have:
Area = (8 * sqrt(231))/2
Area = 4 * sqrt(231)

Therefore, the area of the rhombus is 4 * sqrt(231) square cm.

Conclusion:
The area of the rhombus is 4 * sqrt(231) square cm.