To find the length of side \( a \) in a right-angled triangle where one leg is \( a \), the other leg is \( 4 \) units, and the hypotenuse is \( 5 \) units, we can use the Pythagorean theorem. The theorem states that:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs of the triangle.
Here, we have:
- \( a = a \)
- \( b = 4 \)
- \( c = 5 \)
Using the Pythagorean theorem, we can set up the following equation:
\[ a^2 + 4^2 = 5^2 \]
Calculating the squares:
\[ a^2 + 16 = 25 \]
Next, we isolate \( a^2 \):
\[ a^2 = 25 - 16 \] \[ a^2 = 9 \]
Taking the square root of both sides gives:
\[ a = \sqrt{9} = 3 \]
Thus, the length of side \( a \) is \( \boxed{3} \).