You may not like this, but I feel the need to do it. I’ll derive the equation for the volume of a sphere. *Warning: calculus will be involved here* Let’s examine what a sphere is. Does the proof/explanation without integration possible? Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Section 4-7 : Triple Integrals in Spherical Coordinates. In the previous section we looked at doing integrals in terms of cylindrical coordinates and we now need to take a quick look at doing integrals in terms of spherical coordinates. First, we need to recall just how spherical coordinates are defined. The following sketch shows the ... Volume of hemisphere = Volume of cylinder – volume of inverted cone \ Volume of a sphere = 2 x volume of hemisphere (It is noted that the cross-sectional areas of the solids in both figures may change with different heights from the center of the base. However, this does not affect our proof.) Volume formula derivations Sphere. The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows. The surface area of the circular disk is .