Solution: Let $ r $ be the radius of the forest. The chord length is 14 km, so half is 7 km. The perpendicular distance from the center to the chord is 5 km. Using the Pythagorean Theorem:
["Title: How to Calculate the Radius of a Forest Using Geometry: A Simple Step-by-Step Solution with the Pythagorean Theorem", "Meta Description: Learn how to find the radius of a circular forest using the Pythagorean Theorem—just know the chord length, half the chord, and the perpendicular distance from the center. Step-by-step solution explained.", "---", "### Introduction\nUnderstanding geometric calculations is essential when mapping natural features like forests, lakes, or circular habitats. One common challenge is finding the radius of a circular area when given key measurements such as a chord length and the perpendicular distance from the center to the chord. In this article, we explore a practical application of the Pythagorean Theorem to determine the radius, using a real-world example involving a forest with a 14 km chord.", "---", "### The Problem: Forest Radius Calculation Using Geometry\nImagine a circular forest with radius $ r $. A straight path cuts through the forest forming a chord of length 14 km. The perpendicular distance from the center of the forest to this chord is 5 km—meaning the shortest path from the center to the path is 5 km.", "From geometry, we know:\n- The chord is 14 km long → half the chord is 7 km.\n- The perpendicular from the center to the chord splits the chord into two equal parts.\n- This perpendicular distance (5 km) and half the chord (7 km) form a right triangle with the radius as the hypotenuse.", "---", "### Applying the Pythagorean Theorem\nSince the perpendicular from the center to the chord bisects the chord and meets it at a right angle, we form a right-angled triangle where:\n- One leg is half the chord: $ 7 $ km\n- The other leg is the perpendicular distance: $ 5 $ km\n- The hypotenuse is the radius $ r $", "By the Pythagorean Theorem:\n[\nr^2 = 7^2 + 5^2\n]\n[\nr^2 = 49 + 25 = 74\n]\n[\nr = \sqrt{74}\n]", "---", "### Final Result\nThe radius of the forest is therefore $ \sqrt{74} $ kilometers, which approximates to about 8.6 km.", "---", "### Why This Matters\nThis method is invaluable in environmental mapping, conservation planning, and land surveying. Accurate radius estimation ensures precise land management and efficient resource allocation for forest preservation.", "---", "### Summary\nTo find the radius $ r $ of a circular forest when given:\n- Chord length $ c = 14 $ km → half-chord = $ \frac{c}{2} = 7 $ km\n- Perpendicular distance from center to chord = $ d = 5 $ km\nSimply apply:\n[\nr = \sqrt{\left(\frac{c}{2}\right)^2 + d^2} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74}\n]", "Mastering this geometric approach empowers precise measurements for natural landscapes—serving science, policy, and stewardship.", "---", "Keywords: forest radius calculation, Pythagorean Theorem, chord length formula, geometric measurement, circular area radius, land mapping, forest survey, geometry in nature, environmental data analysis", "Search Intent: Users seeking a clear, step-by-step method to calculate the radius of a circular forest using basic geometry, especially relevant for surveyors, environmentalists, and educators."]
