Mathematical inquiry processes: Interpret; explore; test particular cases. Conceptual field of inquiry: Volume of a pyramid; circumference and perimeter.
The prompt was inspired by the dimensions of the Great Pyramid of Giza, which closely approximate to the condition in the statement. The Ancient Egyptians are believed to have been attempting to 'square the circle' in their construction of the pyramid (see box). There are a number of lines of inquiry that could develop from the prompt:
Define the geometrical terms and understand the meaning of the prompt. The illustration below might help to make it clearer. The semi-circle is folded up in the second picture to show the height of the pyramid equals the radius of the circle.
Draw a net and construct a right square-based pyramid. Work out the perpendicular height of the pyramid using Pythagoras' Theorem and then calculate the circumference of the circle. Compare the perimeter of the square to the circumference of the circle. How closely does the construction satisfy the condition in the prompt? How should we change the dimensions of the net to get closer to satisfying the condition?
Select a side length of the base. Work out the circumference of the circle and the height of the pyramid that satisfy the condition in the prompt as closely as possible.
Calculate the volume of the pyramid and, assuming students selected different side lengths, compare it to the volumes of the mathematically similar pyramids created by other students in the class. What are the scale factors of enlargement for the length and volume in each case? Make a generalisation about the scale factors.
Work out the angles of elevation, the length of the diagonals across the base and the slope heights using Pythagoras' Theorem and trigonometry.
The Great Pyramid of Giza
The Great Pyramid of Piza was completed in 2560 BC. The architects are thought to have based its design on the condition in the prompt. They were attempting to 'square the circle' by finding a square and circle with equal perimeters. At the end of the nineteenth century, 'squaring the circle', which usually refers to the area of the shapes, was proved to be impossible because of the transcendental nature of π.
Attempting to find a circle whose circumference is equal to the perimeter of the square.
The diagram shows a square (side length 1) whose perimeter is 4. The radius of the circle is the perpendicular height of the isosceles triangle. The angles at the base of the triangle are 51.85o, which is the slope angle of each face in the Great Pyramid of Piza. To find the height of the triangle (h):
h = 0.5(tan 51.85o) = 0.6365 (accurate to 4 decimal places).
As the radius is 0.6365, the circumference of the circle = 2πr = 2π(0.6365) = 3.9992 (also accurate to 4 decimal places). The circumference is, therefore, only 0.02% shorter than the length of the perimeter.
The original dimensions of the Great Pyramid are thought to have been:
base length 230.4m height 146.5m
Using these measurements, the perimeter of the base equals 921.6m and the circumference of the circle whose radius is the height of the pyramid is 920.5m (accurate to one decimal place). The circumference is, therefore, 0.12% shorter than the length of the perimeter. A height of 146.66m - just 16cm more - would have given a circumference within 0.012% of the length of the perimeter.
More information about the dimensions of the Great Pyramid of Piza, including their connection to the golden ratio, can be found here.