Circumference and Radius ```Name: Devon H. Status: student Age: N/A Location: N/A Country: N/A Date: N/A ``` Question: I have heard that if a string were tied tightly around the earth's equator it would be 25,000 miles long (roughly). If the string was then cut and six feet was added to the string, and then the string was stretched an equidistant height off the ground, the string would be one foot off the ground. I know this is hypothetical, and involves many generalities (among them the circumference of the earth) but the math seems to play out. 25,000 x 5280 feet / 2pi = radius of earth in feet ((25,000 x 5280 feet)) + 6 feet) / 2pi = new radius If we subtract original radius from new radius we get 1 foot. It does not seem logical. How can I explain this to my students? Replies: Devon, You might try explaining it in general before mentioning the Earth. Circumference equals 2pi times radius. If you add one foot to the radius, circumference increases by 2pi feet. If you add one inch to the radius, circumference increases by 2pi inches. It works for all circles. Another way you may find useful is percentages. Calculate the percent increase for radius. Calculate the percent increase for circumference. You will find them to be equal. Compared to the radius of 21 million feet, one foot is not a large amount. Dr. Ken Mellendorf Physics Instructor Illinois Central College The math is all there in your question. The difficulty is that such a small difference in circumference should not seem to make such a big difference in radius. However, the proportions are preserved. It is true, six additional feet is not much of a difference when compared to the large circumference of the earth. By the same token, though, one additional foot is not much of a difference when compared to the large radius of the earth. Circumference and radius are both linear dimensions, and are proportional to each other. Interestingly, it does not matter what the initial circumference and radius of a circle are, increasing the circumference by six feet will increase the radius by about one foot. You can start from a point (radius = circumference = zero), a circumference of two feet, or the orbit of the earth around the sun. It does not matter. A change of X in the circumference results in a change in the radius of X / (2pi). Richard E. Barrans Jr., Ph.D. PG Research Foundation, Darien, Illinois If you make a relatively small change in the circumference, you would expect to see a relatively small change in the radius, and this is in fact what you see, because the radius changes from 21008452.49 to 21008453.44. Tim Mooney If the circumference of the earth is taken as 25,000x5280 feet the radius is: C/2*pi = 132000000.0 / 2*pi = 2.10084524881302 x 10^7 feet If the circumference is increased to 132000006.0 / 2*pi = 2.10084534430598 x 10^7 feet The increase in radius then is 2.10084534430598 x 10^7 - 2.10084524881302 x 10^7 = 9.54929597618559 x 10^-8 feet. Vince Calder Click here to return to the Mathematics Archives

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