NAME: ABDUL RASYID BIN ABD MANAP

I/C NUMBER: 910715-06-6031

SCHOOL: SEKOLAH MENENGAH SAINS SULTAN HAJI AHMAD SHAH, KUANTAN

ABSTRACT

The value of π is commonly used in finding a value which is related to a circle. For instance, it is used in the formula to find area of circle which is πr², circumference which is 2πr and other formulas. If the constant value of π does not exist, is there any other ways to find the area of circle and its circumference?

This project is mainly about finding the alternative way to find the value of area of circle and its circumference without using the value of π. I combine the formula to find the area of a triangle (½ ab sin θ) with the formula πr² because they are related. This is because if we divide a circle into smaller sectors, each sector will look like a triangle. Therefore, I can combine these two formulas to become a new formula to find the area of circle which is ½ r² sin(1°) x 360° and a new formula to find circumference which is ½ r sin(1°) x 360°.

I compare the value obtained for a circle with radius of 5cm using the two old formulas and the new formulas. By using the old formula to find the area of circle which is πr², the result is 78.53981634 compared to the new formula which is ½ r² sin (1°) x 360°, the result is78.53582897. By using the old formulas to find the circumference of circle which is 2πr, the result is 31.41592654 compared to the new formula which is ½ r sin (1°) x 360°, the result is 31.41433159.

Both new and old formulas had almost similar results but different accuracy. The two new formulas I invented are 99.9% accurate.

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Introduction

The story of pi

Undoubtedly, pi is one of the most famous and most remarkable numbers you have ever met. The number, which is the ratio of circumference of a circle to its diameter, has a long story about its value. Even nowadays supercomputers are used to try and find its decimal expansion to as many places as possible.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter “περίμετρος”, probably by William Jones in 1706, and popularized by Leonhard Eulr some years later. "π" is usually pronounced as pie when used in English in a mathematical context, although the letter is properly pronounced pee in Greek. The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number

It is an irrational number, which means that its decimal expansion never ends or repeats. Indeed: beyond being irrational, it is a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) could ever produce it. Throughout the history of mathematics, much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into culture at large.

Digits of Pi

First 100 digits3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679... First 1000 digits3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

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Circle

The distance around a circle is called its circumference. The distance across a circle through its center is called its diameter. We use the Greek letter (pronounced Pi) to represent the ratio of the circumference of a circle to the diameter. In the last lesson, we learned that the formula for circumference of a circle is: . For simplicity, we use = 3.14. We know from the last lesson that the diameter of a circle is twice as long as the radius. This relationship is expressed in the following formula: .

Area of Circle

First method

the way to find the area of circle = πr²

Second method

The area of a circle is the number of square units inside that circle. If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2.

Title3

Alternative formulas to get the area of circle and its circumference and the discovery of the value of π.

Objective

1) To compare the value of area of circle by using the new formula and the common formula.

2) To compare the value of circumference of a circle by using the new formula and the common formula.

3) To show the new formula to get the value of π which is close to the actual value of π?

Problem statement.

Is there any other formula to get the area of circle and other formula to get the circumference?

Is there any other formula to get the value of π which is close to the actual value of π?

Aim.

To find other formula to get the area of circle, any other formula to get the circumference and to show the new formula to get the value of π which is close to the actual value of π.

Hypothesis.

The value of area of circle and its circumference that we get by using the new formula are quite similar to value that we get by using the old formula.

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Methodology:

1. The area of triangle.

Area of triangle ABC = ½ ab sin C

= ½ ac sin B

= ½ bc sin A

2. The formula of circle.

The formula to find the area of circle = πr²

3. Combining the formula to find the area of circle with the area of triangle

5

ac

b

a

A

B

CѲ

If we divide the circle into smaller sectors, we can see that the shape of the small part is quite similar to the shape of a triangle. If we divide the circle into 360 parts; each of the small sectors will have the angle of 1⁰. By combining the formula to find the area of triangle which is ½ ab sin C with the formula to find the area of circle which is πr², we can get the area of circle without using the value of π.

Thus, the new formula that we can get from both formulas is:

r represents the radius of the circle.

Sin (1°) represents the value of the degree for each sector.

The new formula is derived by simplify the old formula into a simpler form.

4. Testing the new formula and comparing the result obtained from the new and old formula.

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Area of circle= ½ r²sin (1°) x 360

Radius

360 ÷ 2 = 180

Therefore, new formula is

Area of circle= 180 r²sin (1 ° )

No Example Area of circle = πr² New formula = 180 r²sin (1°)

1. Find the area of circle with radius of 7cm.

Area of circle = πr²

= π x 7²

= 153.093804 cm²

Area of circle = 180 r²sin (1°)= 180 x 7²sin (1°)= 153.9302248 cm²

2. Find the area of circle with radius of 9cm.

Area of circle = πr²

= π x 9²

= 254.4690049 cm²

Area of circle = 180 r²sin (1°)= 180 x 9²sin (1°)= 254.4560859 cm²

3. Find the area of circle with radius of 11cm.

Area of circle = πr²

= π x 11²

= 380.1327111 cm

Area of circle = 180 r²sin (1°)

= 180 x 11²sin (1°)= 380.1134122 cm²

Therefore, the values obtained by using both formulas are almost similar. This proved that the new formula can also be used to find the area of circle.

Discussion

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7cm

9cm

11cm

1. Compare the value of π that is obtained.

The new formula which is 180 sin (1°) also allows us get the value of π. The difference between the value of π obtained using the new formula that is 3.141433159 and the value of actual π which is 3.141592654 is 0.00159494. We can get the value of π which is more accurate to the actual value of π by dividing the circle to smaller sectors.

Table below shows the results.

Number of sector of a circle

The degree of each smaller part.

The value of π obtained.

different in values from the formula with the actual value of π

36 10 3.125667189… 0.015925464…

360 1 3.141433159… 0.000159494…

3,600 0.1 3.141591059… 0.000001594…

36,000 0.01 3.141592638… 0.000000015…

360,000 0.001 3.141592653… 0.000000001…

3,600,000 0.000,1 3.141592654… 0.000000000…

Hence, I can conclude that the smaller we divide the circle, the more accurate the value of π obtained by the new formula to find the value of π.

2. Compare the value of pi from the new formula with the other formula.

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In Egypt and in Babylon about two thousand years before Christ. The Egyptians obtained the value (4/3) ^4 and the Babylonians the value 3 1/8 for pi. About the same time, the Indians used the square root of 10 for pi. These approximations to pi had an error only as from the second decimal place.

(4/3)^4 = 3,160493827...

3 1/8 = 3.125

root 10 = 3,16227766...

pi = 3,1415926535...

The next indication of the value of pi occurs in the Bible. It is found in 1 Kings Chapter 7 verse 23, where using the Authorized Version, it is written "... and he made a molten sea, ten cubits from one brim to the other: it was round about ... and a line of thirty cubits did compass it round about." Thus their value of pi was approximately 3. Even though this is not as accurate as values obtained by the Egyptians, Babylonians and Indians, it was good enough for measurements needed at that time.

Jewish rabbinical tradition asserts that there is a much more accurate approximation for pi hidden in the original Hebrew text of the said verse and 2 Chronicles 4:2. In English, the word 'round' is used in both verses. But in the original Hebrew, the words meaning 'round' are different. Now, in Hebrew, Etters of the alphabet represent numbers. Thus the two words represent two numbers. And - wait for this - the ratio of the two numbers represents a very accurate continued fraction representation of pi! Question is, is that a coincidence or ...

Another major step towards a more accurate value of pi was taken when the great Archimedes put his mind to the problem about 250 years before Christ. He developed a method (using inscribed and circumscribed 6-, 12-, 48-, 96-gons) for calculating better and better approximations to the value of pi, and found that 3 10/71 < pi < 3 10/70. Today we often use the latter value 22/7 for work which does not require great accuracy. We use it so often that some people think it is the exact value of pi!

As time went on other people were able come up with better approximations for pi. About 150 AD, Ptolemy of Alexandria (Egypt) gave its value as 377/120 and in about 500 AD the Chinese Tsu Ch'ung-Chi gave the value as 355/113. These are correct to 3 and 6 decimal places respectively.

377/120 = 3, 14166667...

22/7 = 3,142857143...

355/113 = 3, 14159292...

pi = 3,1415926535...It took a long time to prove that it was futile to search for an exact value of pi, ie to show

that it was irrational. This was proved by Lambert in 1761. In 1882, Lindemann proved that

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pi was more than irrational --- it was also transcendental --- that is, it is not the solution of any polynomial equation with integral coefficients. This has a number of consequences

It is not possible to square a circle. In other words, it is not possible to draw (with straight edge, compass and pencil only) a square exactly equal in area to a given circle. This problem was set by the Greeks two thosand years ago and was only put to rest with Lindemann's discovery.

It is not possible to represent pi as an exact expression in surds, like root2, root7 or root5+root3/root7, etc.

From that time on interest in the value of pi has centred on finding the value to as many places as possible and on finding expressions for pi and its approximations, such as these found by the Indian mathematician Ramanujan:

(1 + (root3)/5)*7/3 = 3.14162371...

(81 + (19^2)/22)^(1/4) = 3.141592653...

63(17+15root5)/25(7+15root5) = 3.141592654...

pi = 3.141592654...

According to all the value obtained by the other mathematicians, my value of pi is more accurate compare to the other formula which is,

1,800,000 x sin (0.000,1°) = 3.141592654…

3. Changing the other formula which is related to the circle

If we can change the formula to find area of circle, we also can change the other formula which is related to the circle.

a. Formula to find the circumference of a circle.

Circumference of a circle = 2πr

Substitute the value of π with 180 x sin (1⁰)

The new formula,

No Example Circumference of circle = 2πr Circumference of circle

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2 x 180 x sin (1⁰) x r

=180 x sin (1⁰) x r

1. Find the circumference of circle with radius of 7cm.

Circumference of a circle

= 2πr

= 2 x π x 7

= 43.9822971cm²

circumference of circle = 2 x 180 sin (1⁰) x r= 2 x 180 sin (1⁰) x 7= 43.98006422cm²

2. Find the circumference of circle with radius of 9cm.

Circumference of circle

= 2πr

= 2 x π x 9

= 56.5486677cm²

circumference of circle = 2 x 180 sin (1⁰) x r= 2 x 180 sin (1⁰) x 9= 56.54579686cm²

3. Find the circumference of circle with radius of 11cm.

circumference of circle = 2πr

= 2 x π x 11

= 69.1150383cm²

circumference of circle = 2x 180 sin (1⁰) x r

= 2x 180 sin (1⁰) x 11= 69.11152949cm²

b. Formula to find the area of a sector of a circle.

11

7cm

9cm

11cm

Area of a sector of a circle = A x πr² 360

A represents angle substandard at the center.

r represents the radius of circle.

Substitute the value of π with 180 x sin (1⁰)

The new formula,

No Example Area of a sector of a circle = A x πr² 360⁰

Area of a sector of a circle = A x 180 x sin (1⁰) r²

360⁰

1. Find the circumference of circle with radius of 7cm. Area of a sector of a circle

= A x πr²

360⁰

= 90 ⁰ x π x 7²

360⁰

= 38.48451001cm²

Area of a sector of a circle = A x 180 x sin (1⁰) r²

360⁰ = 90 ⁰ x 180 x sin (1⁰) x 7²

360⁰

= 38.48255619cm²

2. Find the circumference of circle with radius of 9cm. Area of a sector of a circle

= A x πr²

360⁰

= 90 ⁰ x π x 9²

360⁰

= 63.61725124cm²

Area of a sector of a circle = A x 180 x sin (1⁰) r²

360⁰ = 90 ⁰ x 180 x sin (1⁰) x 9²

360⁰

= 63.61402146 cm²

12

7cm

90⁰

9cm

90⁰

A x 180 x sin (1⁰) r²

360⁰

3. Find the circumference of circle with radius of 11cm. Area of a sector of a circle

= A x πr²

360⁰

= 90 ⁰ x πr²

360⁰

= 95.03317777cm²

Area of a sector of a circle = A x 180 x sin (1⁰) r²

360⁰ = 90 ⁰ x 180 x sin (1⁰) x 11²

360⁰

= 95.02835305cm²

Therefore, the value obtained using the new formula is almost the same with the value obtained from the old formula. This proves that the new formula can also be used to find the circumference of a circle and the area of a sector of a circle.

13

11cm

90⁰

Conclusion

The value of area of circle and its circumference that we get by using the new formula are quite similar to value that we get by using the old formula.

Therefore, hypothesis is accepted.

The value of π obtained from 180 sin (1°) is more accurate than the value of π

obtained from other formulas. So, the new formulas can be used to find the area of circle, the circumference of circle

and area of sector.

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Acknowledgement

I would like to take this opportunity to thank the teacher in charge of this project, Miss Ho Chai Ping for her patience and continuous effort in guiding and advising me during the research. With her help and guidance, I managed to complete this research successfully in time.

I would like to thank to Mr. Mohd Yunus Bin Mahil, SM Sains Sultan Hj. Ahmad Shah School’s Vice Principle for his support.

I would like to say thank you to all my friends who are directly or indirectly involved in this research. Thanks to my friend Farhan for his time and help.

Last but not least, this research will not become a reality without my family’s moral support.

………………………………………..

(ABDUL RASYID BIN ABD MANAP)

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Bibliography

1. Http://www .mathgoodies.com/lessons/vol2/circle_area.html

2. http://www.geocities.com/capecanaveral/lab/3550/pi.htm? 200811

3. http://en.wikipedia.org/wiki/pi.

4. http://www.math.com/

5. Chong Pak Cheong. 2006. KBSM. Analysis Series Additional Mathematics SPM. Johor. Penerbitan Pelangi Sdn. Bhd.

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