# Problem Solving

## A Chronological History of Pi with Developmental Activities in Problem Solving

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## Part II: Developmental Activities

### Activity I

The scribe Ahmes assumed that the area of a circular field with a diameter of nine units is the same as the area of a square with a side of eight units.Have students calculate the area of both figures first using the Egyptian ¹. 3 1/6. Than ¹ = 3.14, as known today, and compare the results with the Egyptian’s approximation.

### Activity II

A Babylonian scribe found the area of a circle by taking three times the square of the radius.Have students calculate the area of the circle first using the Babylonian method A = 3r2, and than by today’s method of A = ¹r2.

### Activity III

Heron chose a specific case in which the sum of the circumference, diameter, and area of a circle was 212. Heron’s instructions “Multiply 212 by 154, add 841, take the square root and subtract 29 and divide by 11” will be a circle with diameter 14.Have students read and following Heron’s instructions to calculate the diameter. Is it 14?

### Activity IV

The circumference of a wheel with a diameter of 4 feet is given by Vitruvius as 12.5 feet, implying a value of 3.125 for ¹.Have students using the formula C = ¹d, calculate the value

Now have the students calculate the circumference of the wheel using ¹= 3.14 given a diameter of 4 feet.

Compare: Was Vitruvius off by a great amount?

### Activity V

In the*Nine Chapters on the Mathematical Art*, the area of the circle was found by taking three fourths the square on the diameter or onetwelfth the square of the circumference.

Have students figure the value of ¹ using the following information:

A = (3/4)(d)2 = (1/12)(C)2

¹ should be about 3.

### Activity VI

One statement in the*Aryabhatiya*to which Hindu scholars have pointed with pride is as follows: Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000.

Have students follow the above instructions and calculate ¹.

### Activity VII

V10= 3.162277... is sometimes known as the Hindu value.Have students calculate areas of circles, and circumferences of circles using V10 for ¹.

Compare results with calculations done using 7 ~ 3.14.

### Activity VIII

Using John Wallis’ infinite product:2/¹ = (1/2)(3/2)(3/4)(5/4)(5/6)(7/6) ....

have students calculate ¹ using hand calculators.

How far does one have to carry out the product to approach 3.14?

### Activity IX

Using William Brouncker’s expression:(figure available in print form)

have students calculate ¹ using hand calculators,

How far does one have to carry out the expression to approach 3.14?

### Activity X

Using Leibniz’s infinite series:¹/4 = 1/1 1/3 + 1/5 1/7 + 1/9 ....

have students calculate ¹ using hand calculators.

How far does one have to carry out the infinite series to approach 3.14?

### Activity XI

Using Leonard Euler’s infinite series:¹ = 1 + 1/2 + 1/3 + 1/4 1/5 + 1/6 + 1/7 + 1/8 + 1/9 1/10 + ....

where the sign of a term, after the first two, is determined as follows: If the denominator is a prime of form 4m 1, a minus sign is used, if the denominator is a prime of form 4m + 1, a plus sign is used, and if the denominator is a composite number, the sign indicated by the product of the signs of its components is used.

Have students calculate ¹ using hand calculators.

How far does one have to carry out the infinite product to approach 3.14?

### Activity XII

Have students using hand calculators choose one of these Fourier series (named after Joseph Fourier), to compute ¹ to any desired number of decimal places:/2 = (2/1)(2/3)(4/3)(4/5)(6/5)(6/7) ...

¹2/8 = 1 + 1/9 + 1/25 + 1/49 +...