When you rotate an isosceles right triangle around one of its legs, you form a three-dimensional object known as a cone. In this case, the legs of the triangle serve as the radius and height of the cone.
Since the legs of the triangle measure 4 inches, the radius of the cone will be 4 inches, and the height will also be 4 inches.
The diameter of the base of the cone can be calculated as:
\[ \text{Diameter} = 2 \times \text{Radius} = 2 \times 4 , \text{inches} = 8 , \text{inches} \]
Thus, the diameter of the base of the cone formed is 8 inches.