The hemisphere has radius $ 3r $, so its volume is:
["Understanding the Volume of a Hemisphere with Radius $3r$", "When it comes to calculating the volume of three-dimensional objects, understanding formulas and radii is essential—especially in mathematics, engineering, and architecture. One commonly studied shape is the hemisphere, a half-sphere with a curved surface and a flat circular base. Whether you're designing a dome, modeling a planetary feature, or solving geometry problems, knowing the volume of a hemisphere with radius $3r$ is key.", "### What Is a Hemisphere?", "A hemisphere is exactly half of a full sphere. Its volume is nearly half the volume of a complete sphere, adjusted for the curvature and shape. The formula for the volume $V$ of a full sphere is:", "[\nV = \frac{4}{3} \pi R^3\n]", "Since a hemisphere is half of a sphere, its volume is:", "[\nV = \frac{1}{2} \cdot \frac{4}{3} \pi R^3 = \frac{2}{3} \pi R^3\n]", "In this case, the radius $R$ is given as $3r$. Substituting $3r$ into the formula gives:", "[\nV = \frac{2}{3} \pi (3r)^3\n]", "### Calculating Volume with Radius $3r$", "First, compute $(3r)^3$:", "[\n(3r)^3 = 27r^3\n]", "Now substitute into the volume formula:", "[\nV = \frac{2}{3} \pi \cdot 27r^3 = 18\pi r^3\n]", "### Summary", "So, the volume of a hemisphere with radius $3r$ is:", "[\nV = 18\pi r^3\n]", "This result combines the geometric principle of a hemisphere’s half-sphere volume with algebraic substitution of the scaled radius. Understanding and applying this formula supports accurate calculations in science, engineering, and geometry applications.", "Next time you’re working with hemispheres—whether in education, construction, or design—remember: a hemisphere of radius $3r$ holds a volume of $18\pi r^3$, making precise volume computations both clear and manageable.", "---", "Keywords for SEO: hemisphere volume, volume of hemisphere with radius $3r$, how to calculate hemisphere volume, radius $3r$ Hemisphere, math formula hemisphere volume, 3r hemisphere calculation, $\frac{2}{3} \pi (3r)^3"]
