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Solution: The area of the incircle is $ \pi c^2 $. For a right triangle, the semiperimeter $ s = \frac{a + b + z}{2} $, and the inradius $ c = \frac{a + b - z}{2} $. The area of the triangle is $ r \times s = c \times (z + c) $. Therefore, the ratio is $ \frac{\pi c^2}{c(z + c)} = \frac{\pi c}{z + c} $. The answer is $ \boxed{\dfrac{\pi c}{z + c}} $.
Question: A robotics researcher designs a circular obstacle avoidance path for a robot. If the robot's shortest detour forms a triangle with sides of 7 units, 24 units, and 25 units, what is the radius of the circumscribed circle around its path?
Solution: For a right triangle with legs 7 and 24, and hypotenuse 25, the hypotenuse is the diameter of the circumscribed circle. The radius $ r = \frac{25}{2} = 12.5 $ units. Thus, the radius is $ \boxed{12.5} $.

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