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Write an expression for the length of an arc whose measure is π radians, knowing that the expression for the length of an arc in degrees is two πππ over 360.
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We recall first that an arc is part of the circumference of a circle.
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An arc has a central angle, which we often refer to as π, which is the angle formed by the intersection of the two radii drawn from the endpoints of the arc to the center of the circle.
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Weβre given in the question a formula for calculating an arc length when the central angle of the sector, π, is measured in degrees.
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And we want to find a formula we can use when the angle is instead measured in radians.
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Before we do this, though, letβs remind ourselves where the formula weβve been given comes from.
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We know that the circumference of a full circle is given by the formula ππ or two ππ, where π represents the radius of the circle.
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When weβre finding an arc length though, we only want to find the length of part of the circumference.
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The fraction of the circumference that we have is determined by the central angle π.
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As there are 360 degrees in a full turn, the portion of the circle we have is π over 360.
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So to find the arc length, we multiply the full circumference of the circle by this fraction, giving two ππ multiplied by π over 360, which, of course, is just two πππ over 360 as given in the question.
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We can take two approaches to answering this question then.
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One is to follow the same process from the start but using radians instead of degrees.
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The other is to convert between degrees and radians in our final formula.
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Letβs demonstrate both methods.
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Starting from scratch then, the circumference of a circle as before is two ππ.
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There are two π radians in a full turn.
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So for a central angle π measured in radians, the portion of the circumference that we have is π over two π.
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Multiplying the full circumference of the circle then by this fraction gives two ππ multiplied by π over two π.
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We see that in fact the two π in the numerator will cancel with the two π in the denominator.
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And in this case, the formula for the arc length simplifies to just ππ.
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We multiply the radius π by the central angle π measured in radians.
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The second method, we said, would be to take the formula given in the question for an angle measured in degrees, two πππ over 360, and then convert to radians.
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If we now assume that the angle π in the numerator is measured in radians, we can then replace 360 degrees in the denominator with two π radians.
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That gives two πππ over two π.
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And as before, we see that the factors of two π in the numerator and denominator will cancel each other out, leaving just ππ as before.
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So we find that an expression for the length of an arc whose measure is π radians is ππ.
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We multiply the radius by the central angle π.