Discovering Pi: Circumference, Diameter, and Radius



Discovering Pi: Circumference, Diameter, and RadiusName: What does the circumference of a circle measure?What does the diameter of a circle measure?What does the radius of a circle measure?Highlight the following in different colors:CircumferenceDiameterRadiusWhat is the relationship between the diameter and the radius?Hypothesis:What do you think is the relationship between the circumference and the diameter? (i.e., How many times larger is the circumference than the diameter?)Do you think this is true for all circles?Exploring the Circumference: Diameter RelationshipMeasure the circumference and diameter of circular (and spherical) items in your classroom. All measurements should be double checked for accuracy.Use a calculator to determine the ratio of C ÷ d. Round to the nearest1100or 2 decimal placesItemMeasurement of Circumference (C ) in mmMeasurement of Diameter(d) in mmRatio ofC : d(C ÷ d)1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.A Brief History of PiPi has been known for almost 4000 years – but even if we calculated the number of seconds in those 4000 years and calculated pi to that number of places, we would still only be approximating its actual value. Here’s a brief history of finding pi:ANCIENT BABYLON, 1900 BCE: The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet(circa 1900–1680 BCE) indicates a value of 3.125 for pi, which is a closer approximation.EGYPT, 1650 BCE: In the Egyptian Rhind Papyrus (circa 1650 BCE), there is evidence that the Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for pi.GREECE, 250 BCE: The ancient cultures mentioned above found their approximations by measurement. The first calculation of pi was done by Archimedes of Syracuse (287–212 BCE), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean theorem to find the areas of two regular polygons: the polygon inscribed within7the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, Archimedes showed that pi is between 3 13and10 .71113CHINA, 450 CE: A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method– but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355 . To compute this accuracy for pi, he must havestarted with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places. The Chinese in the 5th century calculated ??to an accuracy not surpassed by Europe until the 1500’s. The Chinese, as well as the Hindus, arrived at ??in roughlythe same method as the Europeans until well into the Renaissance, when Europe finally began to pullahead.THE MAYANS: While pi activity stagnated in Europe, the situation in other parts of the world was quite different. The Mayan civilization, situated on the Yucatan Peninsula in Central America, was quite advanced for its time. The Mayans were top-notch astronomers, developing a very accurate calendar. In order to do this, it would have been necessary for them to have a fairly good value for pi. Though no one knows for sure (nearly all Mayan literature was burned during the Spanish conquest of Mexico), most historians agree that the Mayan value was indeed more accurate than that of the Europeans.THE RENAISSANCE IN EUROPE, 1700 CE: Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Euler, who adopted it in 1737. An 18th century French mathematician named Georges Buffon devised a way to calculate pi based on probability.MODERN DAY: Starting in 1949 with the ENIAC computer, digital systems have been calculating ??to incredible accuracy throughout the second half of the twentieth century. Whereas ENIAC was able to calculate 2,037 digits, the record as of the date of this article is 206,158,430,000 digits, calculated by researchers at the University of Tokyo. It is highly probable that this record will be broken, and there islittle chance that the search for ever more accurate values of ??will ever come to an end.adapted form exploratorium.eduWANT TO KNOW MORE ABOUT Pi?GO TO THIS WEBSITE: with the DiameterExamine the values in your table to determine how the diameter and ??can be used to find thecircumference .Remember ??≈(2 decimal places).State a hypothesis for the circumference formula:C =Test your hypothesis:Measure the diameter (d) of an object in millimetres.Diameter: d =mmCalculate the circumference using ??and the diameter value.Measure the actual circumference of the object.Measured circumference C =mmHow does the calculated circumference compare with the measured circumference?Repeat with a different object.Diameter: d =mmCalculate the circumference, using ? Measured circumference C =mmHow does the calculated circumference compare with the measured circumference?Circus of Circles Calculate the missing measures. Use a calculator. Show your work.a) hoopb) surface of drumc) surface of circular barrelC ??2.5 mr ?d ?d ??75 cmr ?C ?d ??40 cmr ?C ?d) base of tente) hoop waistf)unicycle wheelC ??40 md ?r ?C ??210 cmd ?r ?r ??30 cmd ?C ?The unicycle wheel in part (f) made 1000 complete revolutions as it traveled down a road. What distance did the rider travel?State your answer in units that are more appropriate than centimetres. ................
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