1. Let the three sides of the right triangle, which are in arithmetic progression, be: $a-d,\ a,\ a+d$ where $a+d$ is the largest side, hence the hypotenuse.

2. Since the triangle is right-angled, apply Pythagoras theorem: $(a-d)^2 + a^2 = (a+d)^2$

3. Expand both sides: $a^2 - 2ad + d^2 + a^2 = a^2 + 2ad + d^2$ $2a^2 - 2ad + d^2 = a^2 + 2ad + d^2$

4. Simplify: $a^2 - 4ad = 0$ $a(a-4d)=0$ Since $a>0$, we get: $a=4d$

5. Therefore the sides are: $a-d=4d-d=3d,\quad a=4d,\quad a+d=5d$ So the triangle is a $3d$-$4d$-$5d$ triangle.

6. Area of a right triangle: $\text{Area} = \frac{1}{2} \times (3d) \times (4d) = 6d^2$

7. Given area is $24$: $6d^2 = 24$ $d^2 = 4$ $d = 2$

8. Hence the smallest side is: $3d = 3 \times 2 = 6$

Therefore, the length of the smallest side is $6$.