Eves (1990) presents a proof of the formula as follows. In the figure,
(Click the figure for the GeoGebra applet)
the area of the cross-section of the hemisphere

　the area of the cross-section of the solid at the RHS

Hence the areas of the cross-sections of the two solids at the same height are always equal. By Cavalieri’s principle the volumes of two solids are equal. Hence the volume of the sphere

Ancient Chinese mathematicians approached the problem in a different way. In the classic problem book the Nine Chapters on the Mathematical Art, the volume of a sphere was incorrectly given as π²d³/16 (d = diameter). Chinese mathematician LIU Hui (劉徽, fl. 3rd century) explained the error as follows. It is known that the ratio of the area of a circle inscribed in a square to the area of the square is π/4. So the ratio of the volume of a cylinder inscribed in a cube to the volume of the cube is also π/4, because each cross-section is a circle inscribed in the square and the ratio of their areas is always π/4.

(Click the figure for the GeoGebra applet.)

Now consider also the sphere inscribed in the cube. If we assume, incorrectly, that the ratio of the volume of the sphere to the volume of the cylinder is also π/4, then we would get the volume of the sphere is π²d³/16.

LIU Hui pointed out that the object whose volume is 4/π times that of the sphere should be obtained by replacing each of its cross-section by the circumscribing square (see the figure). He called this object a “double box-lid” (Mouhefanggai 蓋合方蓋). He also pointed out that this object is the intersection of two orthogonal cylinder of the same radius (see the figure).

(Click the figure for the GeoGebra applet.)

LIU Hui was unable to find the volume of the double box-lid. The problem was only solved two centuries later by ZU Gengzhi (祖暅之), the son of the famous mathematician ZU Chongzhi (祖沖之). As shown in the figure below, ZU Gengzhi showed that the volume of the difference between an octant of the double box-lid and the cube containing it is equal to the volume of a pyramid whose volume is one-third of that of the cube. Hence the volume of the double box-lid, and also the volume of the sphere, were found.

(Click the figure for the GeoGebra applet.)

If we try to look at the area of the cross-section of the octant of the double box-lid directly, we could show that its volume is two-third of that of the cube, as shown in the following figures.

         
(Click the figure for the GeoGebra applet.)

Final Remark

It is hoped that the GeoGebra applets presented in these two posts on volumes of pyramids and spheres could help teachers to bridge the “logical gaps” (Tzanakis and Arcavi 2000 p.204-207) in the introduction of these formulas to lower form students without proofs. These materials are also valuable to provide problems to motivate and engage students in the teaching and learning of the formulas.

References

Eves, H. 1990. An introduction to the history of mathematics (Sixth Ed.). Saunders College Publishing

Tzanakis, C., Arcavi, A. 2000. ‘Integrating history of mathematics in the classroom: an analytic survey’, in J. Fauvel and J. van Maanen (eds.), History in Mathematics Education, Kluwer Academic Publishers, 201 – 240

Wagner, D . B. (1978) ‘Liu Hui and Tsu Keng-chih on the volume of a sphere’, Chinese Science Vol.3, 59-79

The following approach of deriving the formula for volume of a pyramid was reported to be originated from Democritus (Edwards 1979, p.8-10).  In the first figure below, a red triangular pyramid is “completed” to form a prism together with the blue and the green pyramids.  Check each of the boxes and drag the blue slider, we can see that the red and the blue pyramids, and also the blue and the green pyramids, always have equal sections at equal heights (the 2nd and the 3rd figures below).  Considering that a pyramid is composed of sections parallel to its base, the red pyramid and the blue pyramid are then equal in volumes, and so are the blue and the green pyramids (Cavalieri’s principle). Since the three pyramids composing the prism are equal in volumes, the volume of the red pyramid is therefore one-third of that of the prism, and hence one-third of its base area times the height.
    
(Click here to access the dynamic figure.)

It is worthwhile to note that a Chinese mathematician LIU Hui (劉徽, fl. 3rd century) proved the result in an ingenious way.  In his commentary of the Chinese’s mathematical classics “Arithmetic in Nine Chapters (九章算術)”, the triangular prism is divided into a triangular pyramid and a rectangular pyramid (the 1st figure below).  After pressing the “Start” and then the “Next” button, with the midpoints of the edges each of the pyramids are divided into some prisms (the red and yellow parts) and some empty smaller pyramids. It is clear that the sum of the volumes of the red parts is half of that of the yellow parts (the 2nd figure below).
 
(Click here to access the dynamic figure.)

The smaller pyramids can be further divided up accordingly (press the “Next” button again to see). Again the sum of the volumes of the red parts is half of that of the yellow parts. The same process can be continued indefinitely (press the “Next” button repeatedly) and we shall see that the limiting shapes, the red triangular pyramid and the yellow rectangular pyramid, have the ratio 1 : 2 in volumes. Therefore the volume of the red triangular pyramid is one-third of that of the prism, and hence the result.
 
 
(Click here to access the dynamic figure.)

LIU Hui expressed his idea of carrying the process to the limit as follows (Wagner, 1979):
The smaller they are halved, the finer are the remaining. The extreme of the fineness is called “subtle”. That which is subtle is without form. When it is explained in this way, why concern oneself with the remainder?
It is particularly impressive that LIU Hui could visualize this subtle limit process, as illustrated by the above dynamic figure, without the help of modern technology.

The above two applets are also available in the GeoGebraBook: http://tube.geogebra.org/material/show/id/359361.

References

Edwards, C. H. Jr. 1979. The Historical Development of the Calculus. Springer-Verlag, New York

Wagner, D.B. 1979. An Early Chinese Derivation of the Volume of a Pyramid: Liu Hui, Third Century A.D. Historia Mathematica 6 p.164-188