How to Use Units of Circular Measurement
Many methods have been used throughout history to measure the arc of a circle, although only two remain in common use, the degree and the radian. The degree probably originated with the ancient Babylonians at least 3,000 years ago. The radian, as a unit of circular measure, is usually credited to Roger Cotes in 1714.
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Instructions
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Use degrees for most practical applications. The Babylonians probably divided a circle into 360 parts because the number 360 has so many factors and because the year was composed of a little more than 360 days. This is still the most widely-used unit of circular measure today.
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Express arcs in radians for mathematical purposes. This is a more natural unit of measure and is used almost exclusively in mathematics. It measures the circumference c of a circle in terms of its radius r. From geometry we know that c = 2 pi r. c/r = 2 pi, so a radian divides a circle into 2 pi, or about 6.28 parts.
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Convert degrees d into radians by multiplying the number of degrees by pi/180. 360 degrees is equal to 2 pi radians so d degrees = 2 pi d/360 radians = pi d/180 radians.
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Calculate the radians r by multiplying the degrees by 180/pi. 2 pi radians = 360 degrees so r radians = 360/2 pi degrees = 180/pi degrees.
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Work with grads under very limited circumstances. This unit is 1/400 of a circle and was introduced as part of the metric system, but it was only adopted in specific areas such as surveying. Today, the grad is only used in the French artillery.
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