After tripling the radius, new radius is \(3r\), so new volume \(V_2\) is:
["Understanding How Volume Changes When Radius Is Tripled: A Clear Guide to Calculating New Volume", "When working with three-dimensional shapes like spheres, cylinders, or cones, the volume depends directly on the radius raised to the third power. If you’re wondering what happens to the volume when the radius is tripled—specifically, when the original radius (r) becomes (3r)—this article explains how to compute the new volume (V_2) step-by-step.", "---", "### The Volume Formula for Common Shapes", "Let’s consider two common shapes where volume depends on radius: the sphere and the right circular cylinder (the principle applies similarly to cones and other radial-symmetric figures).", "For a sphere, the volume is:\n[\nV = \frac{4}{3}\pi r^3\n]", "For a right circular cylinder, volume is:\n[\nV = \pi r^2 h\n]\n(Note: if the height (h) remains constant and the radius increases, this formula helps compute the new volume too.)", "---", "### Step 1: Tripling the Radius", "Suppose the original radius is (r). When tripled, the new radius becomes:\n[\nr_{\ ext{new}} = 3r\n]", "We now compute the new volume (V_2) based on this updated radius. Applying the volume formula:", "- For a sphere:\n[\nV_2 = \frac{4}{3}\pi (3r)^3 = \frac{4}{3}\pi (27r^3) = 27 \cdot \frac{4}{3}\pi r^3\n]\n[\nV_2 = 27V_{\ ext{original}}\n]\nThis means the volume increases by a factor of 27.", "- For a cylinder (height constant):\n[\nV_2 = \pi (3r)^2 h = \pi (9r^2) h = 9\pi r^2 h = 9V_{\ ext{original}}\n]\nHere, the volume increases by a factor of 9.", "---", "### Why Volume Independence by Shape?", "The key insight is that volume scales with the cube of the linear dimension (radius). Since radius is multiplied by 3:\n[\n\ ext{Volume multiplier} = 3^3 = 27\n]\nFor linear dimension changes, volume evolves cubically, unlike area (which squares). This explains why spheres and cylinders both grow 27-fold when radius triples.", "---", "### Real-World Application", "Imagine doubling or tripling the radius of planet orbits, Earth models, or industrial tanks. Instead of estimating volume changes, use this formula to predict precisely how much space increases—critical for engineering, storage planning, or scientific modeling.", "---", "### Final Summary", "- Original radius: (r)\n- New radius: (3r)\n- New volume (V_2):\n - For a sphere: (V_2 = 27 \ imes \frac{4}{3}\pi r^3)\n - For a cylinder (h constant): (V_2 = 9\pir^2h)\n - In any radial-symmetric shape, (V_2 = 27) times the original volume", "---", "Remember: If the height changes in a cylinder, adjust the volume accordingly—but when radius triples and height remains constant, volume scales by 27.", "---", "Target Keywords:\nsphere volume formula, radius tripled volume, volume scaling with radius, cylinder volume change, 3r radius effect, volume calculation step-by-step", "---", "By applying this simple cubic principle, you can accurately determine new volume whenever radius dimensions change—making geometric and engineering calculations fast, accurate, and reliable."]
