Question: A triangular park with sides 7 m, 24 m, and 25 m has an inscribed circle. What is the radius of the circle?
["A Triangular Park with Sides 7 m, 24 m, and 25 m Has an Inscribed Circle—What’s the Radius?", "Curious about hidden geometry in everyday spaces? A growing interest among US residents explores how mathematical properties reveal surprising truths in architecture and parks. One such puzzle centers on a triangular park with side lengths of 7 meters, 24 meters, and 25 meters—known to math enthusiasts for being a Pythagorean triple. But beyond its shape, this triangle holds a deeper secret: it contains a perfectly inscribed circle. Readers often ask: what is the radius of this circle? Understanding this value not only sparks curiosity but also connects to real-world design, sustainability, and smart urban planning.", "Why this triangle has been trending online \nThe promise of a triangle with whole inch-perfect sides—7, 24, 25—resonates in today’s culture. Many users stumble across this puzzle while exploring geometric patterns in public parks or learning about spatial efficiency. Social media discussions and educational videos highlight this triangle as a gateway to practical geometry, blending aesthetics with function. With increased interest in nature integration in city design, identifying the radius helps planners maximize space, improve accessibility, and create child-friendly, circular flow zones within rectangular or triangular parks.", "How math defines the circle inside the triangle \nA triangle’s inscribed circle—formed by touching all three sides internally—hinges on its area and perimeter. For a triangle with sides a, b, and c, the radius r of the inscribed circle is calculated using the formula: \n\[\nr = \frac{A}{s}\n\] \nwhere A is the area and s is the semi-perimeter: \n\[\ns = \frac{a + b + c}{2}\n\] \nFor the triangle with sides 7, 24, and 25, the semi-perimeter is: \n\[\ns = \frac{7 + 24 + 25}{2} = 28\n\] \nSince 7² + 24² = 49 + 576 = 625 = 25², this triangle is right-angled. The area of a right triangle simplifies to: \n\[\nA = \frac{1}{2} \ imes \ ext{leg}_1 \ imes \ ext{leg}_2 = \frac{1}{2} \ imes 7 \ imes 24 = 84\n\] \nThus, the radius becomes: \n\[\nr = \frac{84}{28} = 3\n\] \nThis means the inscribed circle touches each side at one point, with a radius of exactly 3 meters.", "Common questions readers often ask \n- How do you calculate the radius from triangle sides? \nThe formula combines semi-perimeter and area, creating a reliable bridge between geometry and practical measurement. \n- Why is this triangle special for metric parks in the US? \nBecause its integer sides provide clean numerical results—easy for community projects and facility planning. \n- *"]
