The formulas we use in geometry are the results of centuries of study by mathematicians and scientists. The development of a method for determining the value of has a history as part of the "three famous problems" that the Greeks pondered. One of those problems was finding a square whose area is the same as that of a given circle. Early attempts were empirical - that is, experimental - in nature. In the Ahmes Papyrus (circa 1550 b.c.), the area of a circle with diameter d was approximated by a square with sides of length , so the area of the circle was calculated as . If this calculation is compared with our formula for the area of a circle, , it can be shown that the ancient formula is equivalent to a value of = 3.1605. Other methods were also used.

Part I

We will conduct an empirical study similar to those used in ancient times to evaluate . We know that the circumference and diameter of a circle are related by the formula, , or . In order to use this formula to estimate , you will need a long piece of string, a meter stick, and three different circular objects.

1. Wrap a string around the top of the circular object and mark the circumference. Lay the string on a meter         stick and measure the marked length (the circumference).

2. Place the string across the top of the circular object through its center, marking the distance, and measure the string (the diameter).

3. Determine the ratio of the circumference to the diameter or the quotient of the circumference divided by the diameter, rounded to six decimal places.

Complete the following table with your data:
 

Object	Circumference	Diameter	Circumference  diameter

Now determine the average of the three ratios.

The average ratio of the circumference to the diameter is _____________.

The answers in the last column should be close to the same value, as well as to the average of these ratios. In fact, if you measured very accurately, the number should be approximately the value of . Therefore, the real number  is defined as the ratio of the circumference of a circle to its diameter.

Part II

You can research the history of the development of methods for estimating the irrational number in the library and on the Internet. Find a reference that discusses the history of  and write a one-page paper summarizing your findings. Be sure to list your reference source.

Part III

At the beginning of this chapter, there is a photo of the Dome of the Rock. This beautiful structure has a circular dome in its center. Search the Internet to find other photos of the interior and exterior of the structure. Print one of the photos, and include it with your report on this structure. Find information on the measurements of the dome. If you are able to find the measurements, use them to in the formulas for a circle, and apply the formulas to describe the size of the dome. Write a one-page summary of your findings on the beautiful Dome of the Rock.

Part IV

The real number  has many other applications in mathematics. Do research in the library or on the Internet to determine a different application or formula that involves . Write a one-page summary describing the application that uses this mathematical constant. Cite your reference source in your paper.