Introduction:
In this problem, we need to determine whether the triangle with side lengths of 3cm, 4cm, and 5cm is a right-angled triangle or not.

Definition of a Right-Angled Triangle:
A right-angled triangle is a triangle in which one of the angles is a right angle, which measures 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Using the Pythagorean Theorem:
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Applying the Pythagorean Theorem:
Let's calculate the squares of the side lengths of the given triangle:
- The square of the length of the first side (3cm) is 3^2 = 9.
- The square of the length of the second side (4cm) is 4^2 = 16.
- The square of the length of the third side (5cm) is 5^2 = 25.

According to the Pythagorean theorem, if the given triangle is a right-angled triangle, the sum of the squares of the two shorter sides should be equal to the square of the longest side (the hypotenuse).

Checking the Condition:
Let's check if the condition holds true for the given triangle:
- The sum of the squares of the two shorter sides (9 + 16) is 25.
- The square of the longest side (the hypotenuse) is also 25.

Since the sum of the squares of the two shorter sides is equal to the square of the longest side, we can conclude that the given triangle satisfies the Pythagorean theorem and is a right-angled triangle.

Conclusion:
Therefore, the triangle with side lengths of 3cm, 4cm, and 5cm is a right-angled triangle.