Question: A museum curator is analyzing a triangular stained glass panel with side lengths of 7 cm, 24 cm, and 25 cm. What is the radius of the circle circumscribed about the triangle?

Tue, 05 May 2026 14:28:45 +0000

**1. Intro: The Mysterious Geometry Behind a Historic Stained Glass Panel**
Curious visitors and art conservators alike pause when encountering unusual triangles in stained glass—especially one with sides measuring 7 cm, 24 cm, and 25 cm. This shape, formally recognized in geometry, holds a timeless significance: it’s a right triangle, widely celebrated in mathematical patterns and architectural design. Amid growing interest in historical craftsmanship and material preservation, professionals are increasingly analyzing these curves with precision. For a museum curator, understanding the circumscribed circle—also known as the circumcircle—becomes essential. This triangle isn’t just a visual marvel; its geometry reveals deeper stories about design, structural integrity, and cultural context—prompting questions like: *What is the radius of this circle? And why does it matter?* **2. Why This Question Is Gaining Traction in US Museums**
A right triangle with sides 7, 24, and 25 isn’t just a textbook example—it’s increasingly linked to authenticity checks and restoration planning in U.S. collections. As cultural institutions invest in data-driven conservation, precise measurements help determine stress points, balance, and historical alignment. Curators recognize that the circumcircle radius isn’t arbitrary: it informs restoration accuracy and informs how light interacts with the glass, influencing both preservation and visitor experience. This blend of heritage, structure, and science reflects broader trends in museum innovation, making such questions relevant in both academic and public spaces. **3. How to Calculate the Circumradius of a Triangle**
The circumradius \( R \) of any triangle is determined by a simple formula:
\[
R = \frac{abc}{4K}
\]
where \( a \), \( b \), and \( c \) are the side lengths, and \( K \) is the area of the triangle. For a right triangle, this process simplifies. Notice that 7² + 24² = 49 + 576 = 625 = 25², confirming this is indeed a right triangle with 25 cm as the hypotenuse. In right triangles, the circumcenter lies at the midpoint of the hypotenuse, making the circumradius exactly half the hypotenuse. Thus, \( R = \frac{25}{2} = 12.5 \) cm. This elegant result unifies algebra and geometry in a way that resonates strongly with professionals working at the intersection of art and math. **4. Common Questions About the Circumcircle Radius**