Many students never learn about volumes because they do not make it past plane geometry. Those who do often reach calculus by a headlong rush that leaves little or no time for the kind of geometrical thinking on which calculus thrives. Calculus is not the time when students should be doing their first serious thinking about geometry. Rather it should be the culmination of years of consideration of increasingly sophisticated geometrical topics. When a student finally sees the full justification of the formula for the volume of a cone or a sphere, it should be a peak experience, fulfilling a promise implicit in all the experiences he or she has had with cones and spheres all the way through school, beginning in kindergarten.
 
Figure 2. Water in a cylinder exactly fills three cones whose base and height are identical to the base and height or the cylinder. 
Froebel's young students spent a great deal of time pouring water and sifting sand. Differently shaped containers held different amounts, so a student would gradually learn common relationships without even thinking of writing them down. For example, how many conical cups can be filled from the water in a cylindrical cup with the same height and the same base? With a rack of such cups (Figure 2), any student can perform the experiment. The cylinder fills three cups.
We can test this over and over again with different heights and different circular tops. Only later, after the student is familiar with the language of fractions, need this relationship be stated in terms of one volume being onethird of another. Still later, that relationship can be expressed by a formula: the volume of the cone is onethird the area of the base multiplied by the height.
By this time that relationship should already have been observed in other shapes. Three squarebased pyramids can be filled with the sand from one square prism of the same base and height (Figure 3). Even if the base is irregular, this relationship is true. We don't even have to have the center of the cone over the center of the base, assuming that the base even has a center! All this understanding can take place before the student has even seen a fraction, let alone a number like "pi"
 
Figure 3. Water in a prism exactly fills three pyramids whose base and height match those of the prism. This relation holds even for prisms and cones with irregular bases and can be discovered by young children just by pouring water or sand. 
A bit more subtle and even more impressive is the relationship that was symbolized on the gravestone of Archimedes: if a ball fits precisely inside a circular cylinder, then the volume of the ball is twothirds the volume of the cylinder. To illustrate this we can show that three spheres can be filled with the water from two cylinders that encase the spheres (Figure 4). Volumes of irregularly shaped objects can be found by seeing how much water they displace when they are completely submerged. This leads naturally to the notion of density, as a weighttovolume ratio.
 
Figure 4. Pouring water can also verify Archimedes' theorem: the volume of three spheres equals the volume of two cylinders whose radius and height match those of the spheres. 
The notion of area can be introduced by working with volumes. By using a collection of shallow pans, all of the same height, children can compare their volumes and relate them to the areas of their bases. The height dimension is "washed out" if it is the same in all cases. In this way it is easy for children to see that the area of a right triangle is half the area of the associated rectangle and that the area of a scalene triangle is half the area of three different associated parallelograms (Figure 5).
 
Figure 5. By pouring water into shallow pans, children can readily compare the areas of different geometric figures. 
Figure 6. Four right triangles in a square frame reveal a proof of the Pythagorean theorem: the square on the hyupotenuse equals the sum of the squares on the legs of the right triangle. 

We can work as well with tiles of uniform thickness, as Froebel did in his kindergarten gifts in the last century. The relation between the area of a parallelogram and the area of a rectangle can be appreciated at a very early stage by students who actually manipulate physical objects. It isn't necessary to wait until students have learned about square roots before they can see an illustration of the Pythagorean theorem (Figure 6). Children who play with geometric puzzles that illustrate decompositions will find it easier later on to appreciate formal results. [an error occurred while processing this directive]