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# Project kajian matematik pelajar

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### Project kajian matematik pelajar

1. 1. NAME: ABDUL RASYID BIN ABD MANAPI/C NUMBER: 910715-06-6031SCHOOL: SEKOLAH MENENGAH SAINS SULTAN HAJI AHMAD SHAH, KUANTANABSTRACTThe value of π is commonly used in finding a value which is related to a circle. Forinstance, it is used in the formula to find area of circle which is πr², circumference which is2πr and other formulas. If the constant value of π does not exist, is there any other ways tofind the area of circle and its circumference?This project is mainly about finding the alternative way to find the value of area ofcircle and its circumference without using the value of π. I combine the formula to find thearea of a triangle (½ ab sin θ) with the formula πr² because they are related. This is because ifwe divide a circle into smaller sectors, each sector will look like a triangle. Therefore, I cancombine these two formulas to become a new formula to find the area of circle which is ½ r²sin(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 oldformulas and the new formulas. By using the old formula to find the area of circle which isπr², the result is 78.53981634 compared to the new formula which is ½ r² sin (1°) x 360°, theresult is78.53582897. By using the old formulas to find the circumference of circle which is2πr, the result is 31.41592654 compared to the new formula which is ½ r sin (1°) x 360°, theresult is 31.41433159.Both new and old formulas had almost similar results but different accuracy. The twonew formulas I invented are 99.9% accurate.Introduction1
2. 2. The story of piUndoubtedly, pi is one of the most famous and most remarkable numbers you haveever met. The number, which is the ratio of circumference of a circle to its diameter, has along story about its value. Even nowadays supercomputers are used to try and find its decimalexpansion to as many places as possible.The Greek letter π, often spelled out pi in text, was adopted for the number from theGreek word for perimeter “περίμετρος”, probably by William Jones in 1706, and popularizedby Leonhard Eulr some years later. "π" is usually pronounced as pie when used in English ina mathematical context, although the letter is properly pronounced pee in Greek. The constantis occasionally also referred to as the circular constant, Archimedes constant (not to beconfused with an Archimedes number), or Ludolphs numberIt is an irrational number, which means that its decimal expansion never ends orrepeats. Indeed: beyond being irrational, it is a transcendental number, which means that nofinite sequence of algebraic operations on integers (powers, roots, sums, etc.) could everproduce it. Throughout the history of mathematics, much effort has been made to determine πmore accurately and understand its nature; fascination with the number has even carried overinto culture at large.Digits of PiFirst 100 digits3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 59230781640628620899 8628034825 3421170679...First 1000 digits3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 59230781640628620899 8628034825 3421170679 8214808651 3282306647 0938446095 50582231725359408128 4811174502 8410270193 8521105559 6446229489 5493038196 44288109756659334461 2847564823 3786783165 2712019091 4564856692 3460348610 45432664821339360726 0249141273 7245870066 0631558817 4881520920 9628292540 91715364367892590360 0113305305 4882046652 1384146951 9415116094 3305727036 57595919530921861173 8193261179 3105118548 0744623799 6274956735 1885752724 89122793818301194912 9833673362 4406566430 8602139494 6395224737 1907021798 60943702770539217176 2931767523 8467481846 7669405132 0005681271 4526356082 77857713427577896091 7363717872 1468440901 2249534301 4654958537 1050792279 68925892354201995611 2129021960 8640344181 5981362977 4771309960 5187072113 49999998372978049951 0597317328 1609631859 5024459455 3469083026 4252230825 33446850352619311881 7101000313 7838752886 5875332083 8142061717 7669147303 59825349042875546873 1159562863 8823537875 9375195778 1857780532 1712268066 13001927876611195909 2164201989Circle2
3. 3. The distance around a circle is called its circumference. The distance across a circlethrough its center is called its diameter. We use the Greek letter (pronounced Pi) torepresent the ratio of the circumference of a circle to the diameter. In the last lesson, welearned 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 CircleFirst methodthe way to find the area of circle = πr²Second methodThe area of a circle is the number of square units inside that circle. If eachsquare in the circle to the left has an area of 1 cm2, you could count the totalnumber of squares to get the area of this circle. Thus, if there were a total of 28.26squares, the area of this circle would be 28.26 cm2.TitleAlternative formulas to get the area of circle and its circumference and the discovery of thevalue of π.3
4. 4. Objective1) To compare the value of area of circle by using the new formula and the commonformula.2) To compare the value of circumference of a circle by using the new formula and thecommon 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 thecircumference?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 andto 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 arequite similar to value that we get by using the old formula.Methodology:4
5. 5. 1. The area of triangle.Area of triangle ABC = ½ ab sin C= ½ ac sin B= ½ bc sin A2. The formula of circle.The formula to find the area of circle = πr²3. Combining the formula to find the area of circle with the area of triangle5acbaABCѲRadius a
6. 6. If we divide the circle into smaller sectors, we can see that the shape of the small partis quite similar to the shape of a triangle. If we divide the circle into 360 parts; each of thesmall sectors will have the angle of 1⁰. By combining the formula to find the area of trianglewhich is ½ ab sin C with the formula to find the area of circle which is πr², we can get thearea 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.No Example Area of circle = πr² New formula = 180 r²sin (1°)1. Find the area of circle withradius of 7cm.6Area of circle= ½ r²sin (1°) x 360360 ÷ 2 = 180Therefore, new formula isArea of circle= 180 r²sin (1°)
7. 7. Area of circle = πr²= π x 7²= 153.093804 cm²Area of circle = 180 r²sin (1°)= 180 x 7²sin (1°)= 153.9302248 cm²2. Find the area of circle withradius of 9cm.Area of circle = πr²= π x 9²= 254.4690049 cm²Area of circle = 180 r²sin (1°)= 180 x 9²sin (1°)= 254.4560859 cm²3. Find the area of circle withradius of 11cm.Area of circle = πr²= π x 11²= 380.1327111 cmArea of circle = 180 r²sin (1°)= 180 x 11²sin (1°)= 380.1134122 cm²Therefore, the values obtained by using both formulas are almost similar. This proved that thenew formula can also be used to find the area of circle.Discussion1. Compare the value of π that is obtained.The new formula which is 180 sin (1°) also allows us get the value of π. Thedifference between the value of π obtained using the new formula that is 3.141433159 and the77cm9cm11cm
8. 8. value of actual π which is 3.141592654 is 0.00159494. We can get the value of π which ismore accurate to the actual value of π by dividing the circle to smaller sectors.Table below shows the results.Number ofsector of acircleThe degree ofeach smallerpart.The value of πobtained.different in values fromthe formula with the actualvalue of π36 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 πobtained by the new formula to find the value of π.2. Compare the value of pi from the new formula with the other formula.In Egypt and in Babylon about two thousand years before Christ. The Egyptiansobtained 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 asfrom the second decimal place.(4/3)^4 = 3,160493827...3 1/8 = 3.1258
9. 9. root 10 = 3,16227766...pi = 3,1415926535...The next indication of the value of pi occurs in the Bible. It is found in 1 KingsChapter 7 verse 23, where using the Authorized Version, it is written "... and he made amolten sea, ten cubits from one brim to the other: it was round about ... and a line of thirtycubits did compass it round about." Thus their value of pi was approximately 3. Even thoughthis is not as accurate as values obtained by the Egyptians, Babylonians and Indians, it wasgood enough for measurements needed at that time.Jewish rabbinical tradition asserts that there is a much more accurate approximationfor 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 roundare different. Now, in Hebrew, Etters of the alphabet represent numbers. Thus the two wordsrepresent two numbers. And - wait for this - the ratio of the two numbers represents a veryaccurate 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 greatArchimedes put his mind to the problem about 250 years before Christ. He developed amethod (using inscribed and circumscribed 6-, 12-, 48-, 96-gons) for calculating better andbetter approximations to the value of pi, and found that 3 10/71 < pi < 3 10/70. Today weoften use the latter value 22/7 for work which does not require great accuracy. We use it sooften 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 500AD the Chinese Tsu Chung-Chi gave the value as 355/113. These are correct to 3 and 6decimal 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 showthat it was irrational. This was proved by Lambert in 1761. In 1882, Lindemann proved thatpi was more than irrational --- it was also transcendental --- that is, it is not the solution ofany 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 (withstraight edge, compass and pencil only) a square exactly equal in area to a givencircle. This problem was set by the Greeks two thosand years ago and was only put torest with Lindemanns discovery.9
10. 10. • It is not possible to represent pi as an exact expression in surds, like root2, root7 orroot5+root3/root7, etc.From that time on interest in the value of pi has centred on finding the value to as manyplaces as possible and on finding expressions for pi and its approximations, such as thesefound 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 moreaccurate 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 circleIf we can change the formula to find area of circle, we also can change the other formulawhich is related to the circle.a. Formula to find the circumference of a circle.Circumference of a circle = 2πrSubstitute the value of π with 180 x sin (1⁰)The new formula,No Example Circumference of circle = 2πr Circumference of circle=180 x sin (1⁰) x r1. Find the circumference ofcircle with radius of 7cm.Circumference of a circle= 2πr107cm2 x 180 x sin (1⁰) x r
11. 11. = 2 x π x 7= 43.9822971cm²circumference of circle = 2 x 180sin (1⁰) x r= 2 x 180 sin (1⁰) x 7= 43.98006422cm²2. Find the circumference ofcircle with radius of 9cm.Circumference of circle= 2πr= 2 x π x 9= 56.5486677cm²circumference of circle = 2 x 180sin (1⁰) x r= 2 x 180 sin (1⁰) x 9= 56.54579686cm²3. Find the circumference ofcircle with radius of 11cm.circumference of circle = 2πr= 2 x π x 11= 69.1150383cm²circumference of circle = 2x 180sin (1⁰) x r= 2x 180 sin (1⁰) x 11= 69.11152949cm²b. Formula to find the area of a sector of a circle.Area of a sector of a circle = A x πr²360A 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,119cm11cm
12. 12. No Example Area of a sector of a circle= A x πr²360⁰Area of a sector of a circle= A x 180 x sin (1⁰) r²360⁰1. Find the circumference ofcircle with radius of 7cm. Area of a sector of a circle= A x πr²360⁰= 90⁰ x π x 7²360⁰= 38.48451001cm²Area of a sector of a circle= A x 180 x sin (1⁰) r²360⁰= 90⁰ x 180 x sin (1⁰) x 7²360⁰= 38.48255619cm²2. Find the circumference ofcircle with radius of 9cm. Area of a sector of a circle= A x πr²360⁰= 90⁰ x π x 9²360⁰= 63.61725124cm²Area of a sector of a circle= A x 180 x sin (1⁰) r²360⁰= 90⁰ x 180 x sin (1⁰) x 9²360⁰= 63.61402146 cm²3. Find the circumference ofcircle with radius of 11cm. Area of a sector of a circle= A x πr²360⁰= 90⁰ x πr²360⁰= 95.03317777cm²Area of a sector of a circle= A x 180 x sin (1⁰) r²360⁰= 90⁰ x 180 x sin (1⁰) x 11²360⁰= 95.02835305cm²127cm90⁰9cm90⁰11cm90⁰A x 180 x sin (1⁰) r²360⁰
13. 13. Therefore, the value obtained using the new formula is almost the same with the valueobtained from the old formula. This proves that the new formula can also be used to find thecircumference of a circle and the area of a sector of a circle.Conclusion• The value of area of circle and its circumference that we get by using the new formulaare 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 πobtained from other formulas.• So, the new formulas can be used to find the area of circle, the circumference of circleand area of sector.13
14. 14. AcknowledgementI 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 duringthe research. With her help and guidance, I managed to complete this research successfully intime.I would like to thank to Mr. Mohd Yunus Bin Mahil, SM Sains Sultan Hj. AhmadShah School’s Vice Principle for his support.I would like to say thank you to all my friends who are directly or indirectly involvedin 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 moralsupport.14
15. 15. ………………………………………..(ABDUL RASYID BIN ABD MANAP)Bibliography1. Http://www .mathgoodies.com/lessons/vol2/circle_area.html2. http://www.geocities.com/capecanaveral/lab/3550/pi.htm? 2008113. 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.15