Degree Measure of a Central AngleDate: 07/29/97 at 11:38:04 From: Tawny Woods Subject: Trigonometry Find the degree measure, to the nearest tenth, of a central angle whose intercepted arc measures 21 cm. in a circle of radius 4 cm. Date: 08/30/97 at 12:22:42 From: Doctor Sonya Subject: Re: Trigonometry Dear Tawny, To solve this problem, you first need to look at what information you have. All we know is the length of the intercepted arc and the radius of the circle. What else do we need to find to get the answer? If we could somehow figure out what fraction of the circumference the intercepted arc is, wouldn't it make sense to say that the intercepted angle is the same fraction of 360 degrees, the total angular measure the circle? If you draw a picture of a circle with an intercepted arc, this idea becomes a little clearer. To put the above idea into action, we need to figure out the circumference of the circle. There is a very simple formula that says that if a circle has circumference C and diameter D, C = piD where "pi" is a constant equal to 3.14159... with a decimal that goes on forever. Usually I just leave it as "pi" and then plug the value (to however many decimal places I need) at the end. Your calculator might also have a "pi" button that would plug the value in for you automatically. With this formula we can find the circumference of the circle. Say we calculate it to be c. Then the intercepted arc is 21/c of the circle. If c turns out to be, say, 30, then the arc is 21/30ths of the circumference. Now we want to find the central angle that is the same fraction of 360. To find this, call the measure of our central angle A. A/360 will tell us what fraction A is of the total anglular measure. Since we want it it be equal to the first fraction we found, we can write: 21/c = A/360 Once you know what c is, just plug it in, and the value for A shouldn't be too hard to find. What we just did is called setting up a ratio, and is one of the most useful techniques in math, especially in algebra and geometry! -Doctors Sonya and Kelli, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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