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**.

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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°**.

__RADIAN/DEGREE EQUIVALENCES__

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** *

**CHANGING FROM ONE SYSTEM TO THE OTHER:**

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

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 |

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2)

i) 2nd quadrant 135° |
ii) 1st quadrant 72° |
iii) 4th quadrant 1020° |
iv) on negative
– 2700° |
v) 3rd quadrant – 150° |

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3)

a) 57.75^{0} =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 .

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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 - ).*