A circle is inscribed in a triangle with sides measuring 7 cm, 24 cm, and 25 cm. Calculate the area of the inscribed circle.
["# Calculating the Area of the Inscribed Circle in a Triangle with Sides 7 cm, 24 cm, and 25 cm", "A triangle with sides measuring 7 cm, 24 cm, and 25 cm is a right-angled triangle—confirmed by verifying that (7^2 + 24^2 = 25^2). Understanding how to inscribe a circle (incircle) in such a triangle unlocks valuable insights into geometric properties and is highly useful in math, engineering, and design. This article explains how to calculate the area of the inscribed circle, including step-by-step calculations.", "## Understanding the Inscribed Circle", "An incircle is a circle perfectly fitted inside a triangle, tangent to all three sides. Its center, the incenter, is the point where the angle bisectors meet. The radius of the incircle can be found using a natural formula based on the triangle’s area and semiperimeter.", "## Step 1: Verify the Triangle is Right-Angled", "First, confirm that the triangle with sides (a = 7), (b = 24), (c = 25) is right-angled. Since:\n[\n7^2 + 24^2 = 49 + 576 = 625 = 25^2\n]\nthis satisfies the Pythagorean theorem, confirming it’s right-angled at the side combination 7 cm and 24 cm.", "## Step 2: Calculate the Triangle’s Area", "For a right-angled triangle, the area is one-half the product of the perpendicular sides:\n[\n\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{168}{2} = 84 \ ext{ cm}^2\n]", "## Step 3: Calculate the Semiperimeter", "The semiperimeter (s) is half the sum of the triangle’s sides:\n[\ns = \frac{7 + 24 + 25}{2} = \frac{56}{2} = 28 \ ext{ cm}\n]", "## Step 4: Find the Radius of the Inscribed Circle", "The formula connecting the inradius (r), area (A), and semiperimeter (s) is:\n[\nr = \frac{A}{s} = \frac{84}{28} = 3 \ ext{ cm}\n]", "## Step 5: Calculate the Area of the Inscribed Circle", "With radius (r = 3) cm, the area of the incircle is:\n[\n\ ext{Area}_{\ ext{circle}} = \pi r^2 = \pi \ imes 3^2 = 9\pi \ ext{ cm}^2\n]\nUsing the approximate value of (\pi \approx 3.1416),\n[\n9\pi \approx 28.27 \ ext{ cm}^2\n]", "## Conclusion", "In a triangle with sides 7 cm, 24 cm, and 25 cm, the inscribed circle has a radius of 3 cm and an area of (9\pi) cm² (about 28.27 cm² when approximated). Mastering this calculation helps in solving real-world problems involving circles tangent to polygonal boundaries, such as in architecture, manufacturing, and geometry education. Whether you're analyzing triangle properties or designing precision components, understanding incircles enhances accuracy and efficiency.", "Explore perfect circles inscribed in different triangles—each configuration reveals unique mathematical beauty and practical utility.", "---", "This article provides clear insight into inscribing a circle within a specific triangle and calculates the inscribed circle’s area concisely, making it ideal for students, educators, and geometry enthusiasts."]
