Geometric Derivation of Sphere Volume

Can you derive the volume formula for a sphere without calculus integration?

Before Newton and Leibniz formalized calculus in 17th-century Europe, other mathematicians around the world had developed methods to solve geometry problems that set important groundwork. For example, we explored a similar line of reasoning that Archimedes used, the ‘method of exhaustion,’ to compute the area of a circle with an increasing sequence of n-gons. In Euclid’s Elements, he used similar strategies to analyze volume proportions. Another valuable stepping stone towards modern calculus comes from the work of the Italian mathematician Cavalieri – the ‘method of indivisibles.’ For the following activity, we will be using this geometric framework to derive the formula for the volume of a sphere.

The 2D applet below represents a 3D construction that consists of three inscribed solids: a hemisphere (of radius r), a cylinder (with radius = r = height), and a right circular cone with its base matching the top of the cylinder and its apex at the center of the hemisphere base. The applet gives you a side view and a top view.

The dotted line labeled Slice represents a horizontal plane passing through the three solids. As you move it up and down, how do the cross-sectional areas change? Formulate a basic equation that conveys the relationship between the cross-sectional areas of these three objects.

If two objects have equivalent cross-sections for all horizontal slices, what can be said of their volume? (Cavalieri’s Principle) Rewrite the above relationship in terms of volume.

Using the known volume formulas for a cone and a cylinder (remember that radius = height), derive the volume formula for a full sphere.

kbennion, Created with GeoGebra