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Cylinders and Discs

The volume of water in a circular cylinder is a little more than three-quarters of the volume of the rectangular box in which the cylinder just fits (Figure 15). If we pour the water from the cylinder into box-shaped containers of the same height, with square base whose side equals the radius of the cylinder, then we can fill three such boxes and still have some water left over. Experiments with different cylinders and related boxes will quickly show that this pattern works for cylinders of any radius or height. The same ratio, of course, relates the area of a circle to its circumscribing square. Because children can measure poured quantities more easily than painted areas, it may be easier for them to grasp this fundamental ratio first in terms of volume and then subsequently in terms of area.

Figure 15. A set of cups containing a circular cylinder matched to four rectangular boxes of the same height whose bases form a square that encloses the circular base can be used to show that the volume of the cylinder is just a little bit more than the volume of three of the boxes. Hence the area of the circular base is just a bit more than three-quarters of the area of the corresponding square.

The idea of perimeter can be introduced by using a string or a belt, unmarked at first. The distance around a square tile is four times the length of the side of the tile, regardless of the size of the tile. If one circular disc has a radius twice that of another, then a string around the larger will fit twice around the smaller. A string around a disc will go around a square with sides equal to the radius a little more than three times. The crucial fact that the ratio of the circumference of the disc to the perimeter of the square is the same as the ratio of the volume of the cylinder to the volume of the surrounding box would be established only much later. But the fundamental idea that there is a fixed ratio between the perimeter of the disc and the perimeter of a square is something that every child should appreciate, long before any mention of the mysterious number "pi".

The relation between the area and circumference of a circle can be easily seen by cutting a circle like a pie and reassembling the pieces into a nearly rectangular shape. The area of a disc turns out to be equal to the area of a rectangle-like region with one side equal to the radius and the other equal to half the circumference (Figure 16). Subdividing the disc into more slices would make the correspondence even more exact. (Much later students will appreciate the limit concept hidden in this demonstration.) Unfortunately, there seems to be no such nice correspondence between the volume of a sphere and the volume of a rectangular box.

Figure 16. By slicing a circle into thin pie-shaped pieces and reassembling them into a rectangular-shaped region, children can readily see that the area of a circle is the radius (the height of the reassembled rectangle) times half of the circumference (the width of the rectangle).

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