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Pyramid Problems

Many children are fascinated by the great pyramids of Cheops. These only surviving wonders of the ancient world were mathematical challenges to their creators, and they remain challenging today. School study of the monuments of ancient Egypt can be a source of mathematics problems of all sorts, from the most elementary considerations of shadows to the most sophisticated achievement of early mensuration — the volume formula for the frustrum of a truncated pyramid.

Children can decide how to make models of the pyramids. A pile of dry sand or wet sand on a square base provides one example. Models in clay provide another. Students can experiment with different sizes of triangles to see what shapes of pyramids result.

Other monuments of different shapes provide similar exercises in measurement and challenges for construction. What about the burial mounds of American Indians or other cone-shaped structures? What about Mayan pyramids, with their step-like structure? What about Babylonian ziggerats, or pagodas? Each structure provides distinctive features that lead to interesting mathematical questions, which the students themselves can formulate and explore.

A key mathematical notion that arises naturally in the study of monuments is similarity, expressed both algebraically in ratio or proportion and geometrically in shadows and scale diagrams. Consider the following story:

My friend Ambrose sent a snapshot of his trip to Egypt. He is standing next to an obelisk and I can see that his shadow is about one-fourth as long as the shadow of the obelisk. That's a pretty big column, over 24 feet high. I know that because my friend is 6 feet tall. There is a pyramid in the picture too. I can see that its shadow is falling just past the edge of the base. What additional information would I need in order to figure out how high the pyramid is? How can I measure the angle that the slanting side of the pyramid makes with the ground?

Such questions can be discussed at an informal level long before the students deal with triangles formally in geometry and trigonometry.

Thinking about the pyramids can show how problems in different dimensions can illuminate each other. Using the principle of similarity, students can easily calculate the volume of an incomplete pyramid (Figure 12), one of the most important problems in Egyptian mathematics.

Figure 12. An incomplete (or truncated) pyramid poses a challenge to find its volume.

Figure 13. By thinking of a trapezoid as an incomplete triangle, we can find a way to calculate its area that can also be used in three dimensions to find the volume of an incomplete pyramid.

Begin with the analogous problem in the plane: the trapezoid viewed as an incomplete triangle (Figure 13). We know the quantities a, b, and h, and we want to find the area. Assuming that the trapezoid is not a parallelogram, we can complete the figure to a triangle with height that we call x. By observing that the large and small triangles are similar, we see that x/a = (x + h)/b. Hence bx = ax + ah, so x = ha/(b - a) and x + h = hb/(b - a). We then get the familiar formula for area of the trapezoid in a new way, as the difference of the areas of two triangles:

(1/2)(x+h)b - (1/2)xa
=(1/2)hb2/(b-a) - (1/2)ha2/(b-a)
=(1/2)h(b2 - a2) / (b-a)
=(1/2)h(b + a).

The same method enables one to calculate the volume of the incomplete pyramid (Figure 14). We are given the height h of part of the pyramid and the side lengths a and b of the top and bottom squares. If the height of the large pyramid is (x + h), then its total volume will be (1/3)(x + h)b2, while the volume of the small pyramid is (1/3)xa2

Figure 14. By completing the incomplete pyramid, its volume can be calculated as the difference of the volumes of two similar pyramids.

By similar triangles, x/a = (x + h)/b. So, as in the planar case, x = ha/(b - a) and x + h = hb/(b - a). Therefore the volume of the incomplete pyramid is

(1/3)(x+h)b2 - (1/3)xa2
=(1/3)hb3/(b-a) - (1/3)ha3/(b-a)
=(1/3)h(b3 - a3) / (b-a)
=(1/3)h(b2 + ab + a2).

This formula, which was detailed in a papyrus from 1800 B.C., represents a high point in the geometry of the ancient world. It can be appreciated by any student who reaches the level of first-year algebra. Truly enterprising students can conjecture the formula for the volume of an incomplete pyramid in the fourth dimension or in higher dimensions.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.