Solution: To find the radius $r$ of the inscribed circle of a triangle with sides $a = 13$, $b = 14$, and $c = 15$, we use the formula:
["Finding the Radius of the Inscribed Circle: Step-by-Step with Triangle Sides a = 13, b = 14, c = 15", "When working with triangle geometry, one important task is determining the radius of the inscribed circle—the circle that fits perfectly inside the triangle and touches all three sides. This radius, known as the inradius ($r$), plays a key role in various geometric calculations and area determinations. In this article, we explore how to find the inradius of a triangle with known side lengths $a = 13$, $b = 14$, and $c = 15$ using a clear mathematical formula.", "### What is the Inradius?", "The inradius $r$ of a triangle is the distance from the triangle’s incenter (the center of the inscribed circle) to any side—essentially the radius that measures how “deep” the circle fits inside the triangle. It is directly related to the triangle’s area and its semiperimeter, making it easy to compute once those two values are known.", "### Step 1: Compute the Semiperimeter $s$", "The semiperimeter $s$ is half the perimeter of the triangle:", "[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\n]", "### Step 2: Calculate the Area Using Heron’s Formula", "Heron’s formula allows us to compute the area $A$ of a triangle when all three side lengths are known:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Substituting the values:", "[\nA = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Breaking this down:", "[\n21 \ imes 8 = 168,\quad 7 \ imes 6 = 42,\quad 168 \ imes 42 = 7056\n]", "[\nA = \sqrt{7056} = 84\n]", "So the area of the triangle is $A = 84$ square units.", "### Step 3: Use the Inradius Formula", "The inradius $r$ is related to the area and semiperimeter by the formula:", "[\nA = r \ imes s\n]", "Solving for $r$:", "[\nr = \frac{A}{s} = \frac{84}{21} = 4\n]", "### Conclusion", "Therefore, the radius of the inscribed circle of the triangle with sides 13, 14, and 15 is $r = 4$. This elegant geometric solution highlights how combining the semiperimeter and area through Heron’s formula leads directly to the inradius—offering both precision and efficiency for solving real-world and academic geometry problems.", "---", "Key takeaways:\n- Use the semiperimeter to simplify calculations.\n- Heron’s formula enables area computation without needing heights.\n- The inradius $r = \frac{A}{s}$ is the core formula for finding the incircle radius.", "This method applies universally to any triangle — once you know the three side lengths — making it a fundamental tool in triangle geometry and applied mathematics."]
