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A The volume of a Sphere Formula

A sphere is the three-dimensional shape formed by all points on a sphere's surface, which such surface is at every point a plane. The volume of a sphere is calculated by multiplying the density of the sphere by its radius.

The volume of a sphere is the measurement of the space it can occupy. A sphere is a three-dimensional shape that has no edges or vertices. In this short lesson, we will learn to find the volume of a sphere, deduce the formula of volume of a sphere and learn to apply the formulas as well. Once you understand this chapter you will learn to solve problems on the volume of the sphere.

Another approach to obtaining the formula comes from the fact that it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness. (Source: en.wikipedia.org)

$\begingroup$ I think this is the most appropriate answer to the question. There's something about figuring this one out on your own that's very satisfying since you really spend a lot of your life taking this along with several other formulas on faith, but once you see that the derivative of pi*r^2 is the circumference, you start to see why pi isn't just 3.whatever but is truly a ratio that relates a circle's circumference to its diameter. $\endgroup$

Just like before, the radius of a hemisphere is exactly the same as the radius of a sphere. However, when we compare the formula total surface area of a sphere to the surface area formula for a hemisphere, we see that we need to take into account the flat part of the hemisphere. Unlike the volume of a half sphere, the surface area of a half sphere is not exactly half that of the surface area of a sphere. (Source: tutorme.com)