Radians and Arc Length

Degree measure is for angles, but doesn't measure length. We want a system of measurement that allows us to measure not only the rotation of an angle, but also the length of the arc subtended (held up) by that angle. Radians do both.

DEFINITION:

A radian is an angle at the center of a circle which subtends an arc equal to the radius. .

Since each radian cuts off an arc = the radius, if we know how many radii there are in a full circumference we'll know how many radians there are in a full circle.

We know that the full circumference is times the radius.

Since , there must be radians in a full circle.

And therefore, there are radians in half a circle which is a straight line or 180°.          So as you can see, all entries in the table can be obtained

from the basic relationship .

1 radian = 57.29578° or 57.3°

* radians * one system to another * arc length * practice * solutions * Say we wish to find the radian equivalence of 120°.

We use the basic relationship .

Multiplying both sides of the equation by 120, we get:

120° = radians.

However, we know that 120° = 2 times 60°, and from our table we know that ,

so we multiply by 2 to get 120°, which of course gives us .

The same holds true for any multiple of the basic values.

Say we need to find the radian measure of 135° .

We know that 135° is 3 times 45° and we know that ,

so 135° = R .

The same holds true when we need to change from radians to degrees.

Since we know that , we can always replace with 180.

For instance, if we need to know the degree equivalence of radians,
we replace with 180, to get 180 / 15 = 12°.

If there is no in the radian measure, we use the fact that 1 radian = degrees.

Ex: Find the degree measure of 3.5 radians.

Multiply both sides by 3.5 to get:

3.5 R = .

Note: when changing from degrees to radians, leave in the expression.

When going the other way, (from radians to degrees), enter in your calculator (as I did in the previous example), to get a number value.

* radians * one system to another * arc length * practice * solutions * Since the measure of an angle in radians determines the length of the arc subtended by the angle, (it tells us how many radii we've cut off), we can find the length of the arc created by a given rotation or angle if we know the radius of rotation. Ex: Find the length of the arc defined by an angle of 2.4 radians at the center of a circle with radius = 4 cm. Since then L = 2.4 (4) = 9.6 cm.

If the angle is given in degrees, we change it to radians to find the length of the arc.

By the way, this is how the odometer works in a car or on a bicycle.

Should we be given the arc length L and the radius r, we can then find in radians.

* radians * one system to another * arc length * practice * solutions * 1) For the angles listed below:

a) indicate the quadrant in which the angle is found.

b) convert to radian measure. Do not use 3.14 for .

 i) 120° ii) 405° iii) 225° iv) 375° v) – 150°

2) For the angles listed below:

a) indicate the quadrant in which the angle is found

b) convert to degree measure.

 i) ii) iii) iv) v) 3)

a) convert 57.75° to radian measure.

b) convert ( ) radians to degree measure.

c) Find the length of the arc created by a 45° central angle of a circle with a radius of 17 cm.

d) Find the degree measure of the central angle in a circle with radius = 120 cm if it subtends an arc of 132 cm.

e) The wheels of a bicycle have a 24-inch diameter. If the wheels make 12 rotations per minute, how far will the bike travel in 3 minutes? Give the answer in feet to the nearest tenth.

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* radians * one system to another * arc length * practice * solutions * 1)

 i) 120°2nd quadrant ii) 405°1st quadrant iii) 225°3rd quadrant iv) 375°1st quadrant v) – 150°3rd quadrant .

2)

 i) 2nd quadrant135° ii) 1st quadrant72° iii) 4th quadrant1020° iv) on negativex-axis– 2700° v) 3rd quadrant– 150°

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3)

a) 57.750 =57.75 × = 1.00793 R .

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b) ( ) = 12 × (180/7) = 308.57 o .

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c) Find the length of the arc created by a 45° central angle of a circle with a radius of 17 cm. , so = 13.35 cm

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d) Find the degree measure of the central angle in a circle with radius = 120 cm if it subtends an arc of 132 cm.

Since , with in radians -- we need only divide both sides by r to get .  .

e) The wheels of a bicycle have a 24-inch diameter. If the wheels make 12 rotations per minute, how far will the bike travel in 3 minutes? Give the answer in feet to the nearest tenth.

Since the diameter = 2 ft., the radius = 1 ft. (24 inches = 2 feet for metric users)

Since 1 rotation/min = radians/min, 12 rotations/min = radians/min.

Therefore, in 3 minutes the wheel travels through radians.

So, the bike will travel feet or 226.2 feet.

* radians * one system to another * arc length * practice * solutions * (all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).