The radius \( r \) of the inscribed circle is given by \( r = rac{A}{s} = rac{60}{20} = 3 \).

["Understanding the Radius of the Inscribed Circle: The Formula and Application", "In geometry, one of the fascinating relationships involving special triangles centers around the radius ( r ) of the inscribed circle (also known as the incircle). This radius plays a crucial role in various mathematical problems, from calculating areas to understanding the properties of triangles. This article explores the derivation and significance of the formula ( r = \frac{A}{s} ), particularly focusing on how it allows us to find the inradius when the area ( A ) and semi-perimeter ( s ) of a triangle are known.", "---", "### The Formula: ( r = \frac{A}{s} )", "The radius ( r ) of the inscribed circle in any triangle can be computed using the elegant formula:", "[\nr = \frac{A}{s}\n]", "Where:\n- ( r ) is the radius of the incircle\n- ( A ) is the area of the triangle\n- ( s ) is the semi-perimeter of the triangle, defined as ( s = \frac{a + b + c}{2} ), with ( a, b, c ) being the lengths of the triangle’s sides.", "---", "### Step-by-Step Explanation", "Let’s break down how this formula arises and how you can apply it effectively.", "1. Understanding the Semi-Perimeter ( s ):\nThe semi-perimeter ( s ) is half the perimeter of the triangle. For a triangle with sides ( a ), ( b ), and ( c ):", "[\ns = \frac{a + b + c}{2}\n]", "2. Area Connection Through the Inradius:\nThe area ( A ) of a triangle with an incircle touches all three sides and can be expressed as:", "[\nA = r \ imes s\n]", "This equation comes from summing the areas of three smaller triangles formed by connecting the incenter (center of the incircle) to the vertices. Each small triangle has base as a side of the original triangle and height ( r ), so:", "[\nA = \frac{1}{2} a r + \frac{1}{2} b r + \frac{1}{2} c r = \frac{1}{2} r (a + b + c) = r \ imes s\n]", "3. Solving for ( r ):\nRearranging the area formula, we solve for ( r ):", "[\nr = \frac{A}{s}\n]", "---", "### Example Calculation", "Consider a triangle with area ( A = 60 ) and semi-perimeter ( s = 20 ). The radius of the inscribed circle is found by:", "[\nr = \frac{A}{s} = \frac{60}{20} = 3\n]", "This means the incircle of the triangle has a radius of 3 units — a concise and powerful relationship linking area, perimeter, and inradius.", "---", "### Why This Formula Matters", "- Efficient Computation: When you know the triangle’s area and sides (or semi-perimeter), this formula eliminates the need for complex inradius constructions.\n- Practical Applications: This concept is useful in architecture, engineering, and design where circular fittings must fit perfectly inside polygonal enclosures.\n- Theoretical Insight: It connects fundamental triangle properties — area, perimeter, and incircle radius — revealing the deep symmetry within Euclidean geometry.", "---", "### Real-world Significance", "Imagine designing a circular garden inside a triangular plot. Knowing the triangle’s area and semi-perimeter allows you quickly to determine the maximum size of the circle that fits snugly within the plot. Using ( r = \frac{A}{s} ), you efficiently calculate the usable radius, optimizing space and design.", "---", "### Summary", "The formula ( r = \frac{A}{s} ) provides a straightforward, powerful way to determine the radius of the inscribed circle:", "- Use the triangle’s area ( A ) and semi-perimeter ( s )\n- Divide ( A ) by ( s ) to find ( r )\n- This relationship simplifies geometric computations and enhances understanding of triangle properties", "This elegant connection underscores the harmony in geometry — where area, perimeter, and incircle radius interrelate through a single, clear equation.", "---", "Key takeaways:\n- Formula: ( r = \frac{A}{s} )\n- Semi-perimeter: ( s = \frac{a + b + c}{2} )\n- Area = semi-perimeter × inradius\n- Useful for exact calculations and real-world applications", "Whether in math class, architectural design, or engineering, mastering this formula empowers you to unlock essential geometric relationships with ease.", "---", "Keywords: inscribed circle radius, inradius formula, geometric formula, area semi-perimeter, triangle incircle, ( r = \frac{A}{s} ), geometric calculation"]