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} $.

["Understanding the Circumscribed Circle of a Right Triangle: Radius Formula and Calculation", "In geometry, one of the most elegant relationships in right triangles is that the hypotenuse is always the diameter of the circumscribed circle. This principle makes both visualizing and calculating key circle properties simple and intuitive—especially when working with a 7-24-25 right triangle.", "### The Right Triangle with Legs 7 and 24, Hypotenuse 25", "Consider a right triangle with legs measuring 7 units and 24 units. Using the Pythagorean theorem:", "[\na^2 + b^2 = c^2 \Rightarrow 7^2 + 24^2 = 49 + 576 = 625 = 25^2\n]", "This confirms the hypotenuse is correctly ( c = 25 ), making it a classic example of a Pythagorean triple.", "### Hypotenuse as Diameter of the Circumscribed Circle", "A fundamental theorem in geometry states that for any right triangle, the circumcircle—the circle passing through all three vertices—has the hypotenuse as its diameter. Because the hypotenuse serves as the diameter, the radius ( r ) of the circle is simply half the length of the hypotenuse:", "[\nr = \frac{\ ext{hypotenuse}}{2} = \frac{25}{2} = 12.5\n]", "### Bridge Between Triangle and Circle", "This relationship gives a straightforward way to compute the radius of the circumscribed circle using only the triangle’s sides:", "[\n\boxed{r = 12.5}\n]", "### Why This Matters", "Recognizing that the hypotenuse equals the circle’s diameter offers more than a formula—it enables quick calculations, deeper geometric insight, and applications in construction, design, and trigonometry. Whether solving problems or appreciating symmetry in shapes, this property remains a cornerstone of right triangle geometry.", "---", "Summary\n- Right triangle with legs 7, 24, hypotenuse 25\n- Hypotenuse = diameter ⇒ radius = ( \frac{25}{2} )\n- Circumscribed circle radius = ( \boxed{12.5} ) units", "Leverage this key insight to simplify your geometric reasoning and calculations!"]