Das Volumen der Kugel beträgt (4/3)πr³ = (4/3)π(4³) = (256/3)π cm³.
["The Volume of a Sphere: Understanding the Formula and Calculation", "When it comes to geometry, few shapes are as fundamental as the sphere. Whether it’s a planet, a marble, or a bubble, understanding its volume helps in everything from engineering to everyday applications. One frequently encountered formula is:", "> The volume of a sphere = (4/3)πr³", "But what does this really mean, and how is it calculated? In this article, we break down the calculation step-by-step, including a practical example using the radius of 4 cm.", "---", "### What is the Volume of a Sphere?", "The volume of a three-dimensional sphere measures the amount of space it occupies. Unlike surface area, which represents how much area the surface covers, volume quantifies the internal capacity of the shape.", "The general formula for the volume of a sphere is:\nV = (4/3)πr³\nwhere\n- V = volume\n- r = radius of the sphere\n- π (pi) ≈ 3.14159", "---", "### Why Is It (4/3)πr³?", "The derivation of this formula comes from advanced geometry and calculus, but intuitively, it reflects how adding layers increases volume as the cube of the radius — scaled by a proportional constant. For a sphere of radius 4 cm, this formula allows us to compute its total internal volume with precision.", "---", "### Step-by-Step Calculation: Volume with r = 4 cm", "Let’s apply the formula to a sphere with radius r = 4 cm:", "1. Start with the formula:\n [\n V = \frac{4}{3} \pi r^3\n ]", "2. Substitute r = 4 into the formula:\n [\n V = \frac{4}{3} \pi (4)^3\n ]", "3. Calculate the cube of the radius:\n [\n 4^3 = 64\n ]", "4. Multiply by π and the constant:\n [\n V = \frac{4}{3} \pi \ imes 64 = \frac{256}{3} \pi\n ]", "5. Compute the numerical value (optional):\n Using π ≈ 3.14159,\n [\n V ≈ \frac{256}{3} \ imes 3.14159 ≈ 85.333 \ imes 3.14159 ≈ 268.08 \ ext{ cm}^3\n ]\n But the exact value remains (256/3)π cm³, preferred in mathematical precision.", "---", "### Practical Applications of Sphere Volume", "Knowing the volume helps in:", "- Designing spherical containers or tanks\n- Understanding bell domes and celestial bodies\n- Calculating drug dosages based on spherical dosage forms\n- Analyzing fluid displacement in spheres", "---", "### Conclusion", "The formula V = (4/3)πr³ is a cornerstone of geometry, offering a reliable way to compute the volume of any sphere. Using a radius of 4 cm, we find the volume equals (256/3)π cm³ — an exact expression ideal for scientific and engineering work. Whether studying physics, chemistry, or architecture, mastering this concept empowers precise calculations and deeper understanding of spatial properties.", "---", "Keywords: sphere volume, volume of a sphere formula, (4/3)πr³, calculus in geometry, spherical volume calculation, πr³, 4 cm sphere, geometry tutorial, exact volume formula."]
