Dividing a Circle into 8 Equal Parts
Date: 08/19/2002 at 22:23:17 From: Thomas Weir Subject: Dividing a circle into equal parts If you have a diameter (= 14ft), can you mark off the 1/8 sides of the circumference so that they are equal? Thank you, Tom P.S. I'm a contractor, and the concrete is coming at 1:00 P.M. tomorrow.
Date: 08/20/2002 at 09:15:49 From: Doctor Rick Subject: Re: Dividing a circle into equal parts Hi, Tom. We're glad to be of help to people who need math in their real-world jobs! Many students ask us, "When will I ever use this stuff?" Are you saying that the circle with diameter 14 feet is marked out, and all you need to do now is divide the circumference into 8 equal segments? The most direct method would be to use a tape measure to lay out a length of 1/8 the circumference along the arc. Since the circumference of a circle is pi (about 3.1416) times the diameter), this distance is 1/8 * 3.1416 * 14 feet = 5.498 feet = 5 feet, 5.97 inches Measuring along a curved line isn't easy, so we can do better. We can find the *straight-line* distance between the marks on the circle. Imagine that we have the marks and we connect opposite marks with diameters of the circle. There will be 4 lines, and each will make an angle of 45 degrees with its neighbors. Now connect the neighboring marks on the circumference; we have made 8 isosceles triangles (two sides the same length, equal to the radius of the circle). We want to find the length of the third side of this triangle, knowing the other sides and the angle. I won't go into the trigonometry involved in the next step unless you'd like to see it. The answer is: L = D*sin(45/2 degrees) where L is the distance we seek and D is the diameter. Using a calculator to find the sine of 45/2 = 22.5 degrees, I get: L = D*0.382683 With a diameter of 14 feet, L = 5.3576 feet, or 5 feet, 4.29 inches. Notice that this is a bit less than the distance around the curve: a straight line is the shortest distance between two points. The upshot is this: Make the first mark on the circle, then measure a straight-line distance of 5 feet, 4.29 inches from there to find the next point on the circle. You could keep going this way all the way around (probably ending up with a little error at the end); or if you know where the center of the circle is, you might make just 4 marks, then extend diameters from these 4 points through the center of the circle to find the other four points. You probably know better than I how to minimize the errors in measurement. (My house projects don't come out that well!) - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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