Question: A cartographer uses satellite data to map a circular forest reserve. A straight river crosses the reserve forming a chord of length 14 km, and the shortest distance from the forest’s center to the river is 5 km. What is the area of the forest in square kilometers?

["How Satellite Mapping Reveals the Area of a Circular Forest Reserve Using a Crossing River", "When cartographers use satellite data to map natural features like forest reserves, precise geometric analysis becomes essential—especially when natural formations such as rivers form chords across circular areas. A recent mapping project demonstrates this expertise using a circular forest reserve intersected by a straight river. By analyzing the river’s chord length and its shortest distance from the forest’s center, we can accurately determine the forest’s total area.", "### The Geometry Behind the Map", "In this scenario, the forest reserve is circular. A straight river cuts across the reserve, forming a chord of length 14 km. The shortest distance from the center of the forest to the river is 5 km—a critical metric in an application of the Pythagorean Theorem in circle geometry.", "Let’s break down the key elements:", "- Chord length (L) = 14 km\n- Distance from center to chord (d) = 5 km", "This distance $ d $ represents the perpendicular from the center $ O $ of the circle to the chord. Since this perpendicular bisects the chord, it divides the chord into two segments of length $ \frac{L}{2} = 7 $ km each.", "Together, $ d $, $ \frac{L}{2} $, and the radius $ r $ of the circle form a right triangle, with $ r $ as the hypotenuse:", "$$\nr^2 = d^2 + \left(\frac{L}{2}\right)^2\n$$", "Substituting the known values:", "$$\nr^2 = 5^2 + 7^2 = 25 + 49 = 74\n$$", "Therefore, the radius of the forest reserve is:", "$$\nr = \sqrt{74} \ ext{ km}\n$$", "The area $ A $ of a circle is given by $ A = \pi r^2 $. Since $ r^2 = 74 $, the area is:", "$$\nA = \pi \ imes 74 = 74\pi \ ext{ square kilometers}\n$$", "Approximating $ \pi \approx 3.1416 $, the area is about 232.24 square kilometers—though the exact value remains $ 74\pi $.", "### Why This Precision Matters in Satellite Mapping", "Accurately determining the area of protected regions like forest reserves relies heavily on integrating satellite imagery with fundamental geometric principles. For conservation planning, environmental monitoring, and land-use policy, knowing the exact size of a circular reserve ensures proper management, tracking of deforestation, and efficient allocation of resources.", "By measuring the river’s chord and its offset from the center, cartographers don’t just map lines—they decode the shape of natural ecosystems, enabling smarter, data-driven decisions.", "In summary, satellite data combined with geometric analysis empowers accurate mapping of the forest’s circular boundary. The river’s chord and proximity to the center unlock the circle’s area through simple yet powerful math—proving that even ancient cartography evolves with modern precision.", "---", "Key Terms: Circular forest reserve, river chord map, cartography, satellite imagery, Pythagorean Theorem, area calculation, geometric analysis, forest reserve geometry, Oak-forest reserve mapping.\nMeta Keywords: how to calculate forest area with satellite data, circular reserve area formula, river chord and circle radius, forest boundary mapping, protected area geometry"]