Question: A marine safety consultant studies a triangular coral reef formation with sides of 10 m, 10 m, and 12 m. What is the radius of the inscribed circle in meters?
["Understanding the Radius of the Inscribed Circle in a Triangular Coral Reef Formation", "When a marine safety consultant examines a triangular coral reef with sides measuring 10 meters, 10 meters, and 12 meters, a key geometric question arises: What is the radius of the inscribed circle? This measure plays an essential role in assessing reef stability, habitat coverage, and ecological resilience—important factors in marine safety and conservation planning.", "In geometry, the inscribed circle (or incircle) of a triangle is the largest circle that fits perfectly inside the triangle, touching all three sides. Its radius can be calculated using the formula:", "[\nr = \frac{A}{s}\n]", "where:\n- ( r ) is the radius of the inscribed circle,\n- ( A ) is the area of the triangle,\n- ( s ) is the semi-perimeter of the triangle.", "### Step 1: Calculate the Semi-Perimeter", "Given the side lengths ( a = 10 , \ ext{m} ), ( b = 10 , \ ext{m} ), and ( c = 12 , \ ext{m} ), the semi-perimeter ( s ) is:", "[\ns = \frac{a + b + c}{2} = \frac{10 + 10 + 12}{2} = \frac{32}{2} = 16 , \ ext{m}\n]", "### Step 2: Compute the Area Using Heron’s Formula", "Heron’s formula gives the area ( A ) of a triangle based on its side lengths:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Substituting values:", "[\nA = \sqrt{16 \ imes (16 - 10) \ imes (16 - 10) \ imes (16 - 12)} = \sqrt{16 \ imes 6 \ imes 6 \ imes 4}\n]", "[\nA = \sqrt{16 \ imes 36 \ imes 4} = \sqrt{2304} = 48 , \ ext{m}^2\n]", "### Step 3: Calculate the Inscribed Circle Radius", "Now substitute ( A = 48 , \ ext{m}^2 ) and ( s = 16 , \ ext{m} ) into the radius formula:", "[\nr = \frac{A}{s} = \frac{48}{16} = 3 , \ ext{m}\n]", "### Conclusion", "For the triangular coral reef formation with sides 10 m, 10 m, and 12 m, the radius of the inscribed circle is 3 meters. This information helps marine consultants assess the spatial dynamics of reef structures, supporting safer navigation, habitat conservation, and effective risk management in coastal and marine environments.", "Understanding such geometric properties enhances environmental stewardship and aids in designing protective measures for vulnerable reef ecosystems under increasing anthropogenic and climatic pressures.", "---", "Key Takeaways:", "- The triangle is isosceles with sides 10 m, 10 m, 12 m.\n- The inscribed circle radius is 3 meters.\n- The formula ( r = A / s ) efficiently combines area and semi-perimeter.\n- This calculation supports marine safety and ecological monitoring.", "Keywords: geometric reef analysis, inscribed circle radius, marine safety, coral reef geometry, Heron’s formula, triangle inradius, coral habitat protection"]
