Problem 1 (Aztec Pyramid)

Consider an Aztec pyramid that has a square bottom and square top (like an Egyptian pyramid but with a square top). The base of the pyramid is a square with dimensions 81 by 81 meters, and the top is a square with dimensions 16 by 16 meters. The four edges that run up the sides of the pyramid are each 65 meters long. We wish to design a stairway that starts on the ground at one of the four corners of the pyramid and corkscrews up and around the four sides in such a way that it goes up at a constant rate until it reaches the top on the same edge above where it started. How high up along the edges of the pyramid are the points where the stairs goes around the three corners eventually reaching the top at a height of 65 meters ?

Solution

The following diagram shows the strairs moves up the faces of the four faces of the pyramid showing the points S₁, S₂, S₃, and D where the stairs goes around the corners. The point D is the top point.

A simple calculation shows us that the base angle of the pyramd is 60° . If we also observe that the triangles ABS'₁, S₂S'₂ S₃, and S'₃S₃D in the diagram are similar (they have equal angles), we have the following properties: From these values, we can find the four differences

• A B - S₁S' = 81 - 54 = 27
• S₁ S'₁ - S₂ S'₂ = 54 - 36 = 18
• S₂ S'₂ - S₃ S'₃ = 36 - 24 = 12
• S₃ S'₃ - C D = 24 - 16 = 8

from which we can use the basic properties of a 30-60 right triangle to conclude

• B S₁ = 27
• S₁ S₂ = 18
• S'₂ S₃ = 12
• S₃ D = 8

Which is a total of 65 meters.

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Last modified on Tuesday, January 12, 1999