Question:** An environmental engineer is assessing a triangular plot of land with side lengths 7 m, 24 m, and 25 m, which is to be used for a rainwater collection system. What is the radius of the inscribed circle within this triangle?
["How to Calculate the Inradius of a Triangle: A Practical Example for Rainwater Harvesting Systems", "When designing sustainable rainwater collection systems, engineering precision is essential—especially when selecting the optimal site for collection infrastructure. One key geometric consideration is the inradius (radius of the inscribed circle) of the land plot. For engineers working with triangular land parcels, understanding this value helps determine efficient space utilization and structural planning.", "In this article, we explore how to calculate the inradius of a triangular plot with side lengths 7 m, 24 m, and 25 m—perfect for a rainwater harvesting system due to its strong geometric properties.", "### Why Environmental Engineers Use Triangles for Rainwater Systems", "Triangular plots are often ideal for rainwater collection because:", "- They offer defined boundary perimeters and areas that can be easily calculated.\n- The inscribed circle’s radius helps maximize catchment efficiency by ensuring optimal catchment-to-storage ratios.\n- Triangular layouts support modular overflow and drainage design.", "Using real-world data, let’s determine the inradius of a triangle with side lengths:\na = 7 m, b = 24 m, c = 25 m", "---", "### Step 1: Verify the Triangle Type", "Before calculating the inradius, confirm whether the triangle is valid and type, specifically right-angled.", "Check using the Pythagorean theorem:", "[\n7^2 + 24^2 = 49 + 576 = 625 = 25^2\n]", "Since this holds true, the triangle is a right triangle with the right angle between the sides of 7 m and 24 m, and hypotenuse 25 m.", "---", "### Step 2: Calculate Area of the Triangle", "For a right triangle, the area is simply:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{leg 1} \ imes \ ext{leg 2} = \frac{1}{2} \ imes 7 \ imes 24 = 84 \ ext{ m}^2\n]", "---", "### Step 3: Compute Semi-Perimeter", "The semi-perimeter ( s ) is:", "[\ns = \frac{a + b + c}{2} = \frac{7 + 24 + 25}{2} = \frac{56}{2} = 28 \ ext{ m}\n]", "---", "### Step 4: Calculate Inradius ( r )", "The inradius ( r ) of any triangle is given by:", "[\nr = \frac{\ ext{Area}}{s}\n]", "Substitute the known values:", "[\nr = \frac{84}{28} = 3 \ ext{ m}\n]", "---", "### Why This Radius Matters for Rainwater Systems", "An inradius of 3 meters means that an inscribed circular catchment area (possibly with piping and filtration) can be centrally placed to efficiently channel runoff toward storage tanks located at the triangle’s incenter—the point equidistant from all sides. For engineers, this precise measurement enables optimal positioning of infrastructure and ensures minimal dead zones in water collection.", "---", "### Summary", "- Triangle with sides 7 m, 24 m, 25 m is a right triangle.\n- Area = 84 m²\n- Semi-perimeter = 28 m\n- Inradius = Area ÷ Semi-perimeter = 3 m", "For environmental engineers assessing triangular plots, knowing the inradius supports smart, sustainable design—especially in rainwater harvesting applications where every square meter and centimeter counts.", "---", "Keywords: environmental engineer, rainwater collection, inradius calculation, triangle inscribed circle, right triangle, sustainable infrastructure, catchment area radius, environmental design, triangle geometry, water harvesting systems.", "---", "Explore more: Use accurate geometric calculations like the inradius to maximize efficiency in green engineering projects."]
