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The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. In geometry, both degrees and radians can be used to represent the measure of an angle.

When measuring angles in degrees, a measure of one degree $\left(1\xb0\right)$ is equivalent to a rotation of $\frac{1}{360}$ of a complete revolution. Half of a revolution is $180\xb0$ .

Radians are another way to measure an angle. A full revolution in radians is 2 radians. Half a revolution is radians. Radians are often a good choice for angles because they are a dimensionless unit.

So we now know that degree and radian measure must be related by the equations

$360\xb0=2\pi \mathrm{radians}$

And

$180\xb0=\pi \mathrm{radians}$

The value of $180\xb0$ is equal to $\pi $ radians. Therefore, to convert any given angle from the measure of degrees to radians, the value has to be multiplied by $\frac{\pi}{180\xb0}$ .

$\mathrm{Angle\; in\; radian}=\mathrm{Angle\; in\; degree}\times \frac{\pi}{180\xb0}$

Where the value of π is approximately 3.14.

- The following steps are used to convert degrees to radians:
- Write the numerical value of an angle given in degrees.
- Now multiply the numerical value written in step 1 by $\frac{\pi}{180\xb0}$ .
- Simplify the expression by canceling the common factors of the numerical.
- The result obtained after the simplification will be the angle measured in radians.

**Example 1**

Convert $90\xb0$ to radians.

$90\xb0=90\times \frac{\pi}{180}\mathrm{radians}$

$90\xb0=\frac{\pi}{2}\mathrm{radians}$ or 1.571 radians

**Example 2**

Convert $60\xb0$ to radian measure.

$=60\times \frac{\pi}{180}\mathrm{radians}$

$=\frac{\pi}{3}\mathrm{radians}$ or 1.047 radians

**Example 3**

Convert $150\xb0$ to radian measure.

$=150\xb0\times \frac{\pi}{180\xb0}\mathrm{radians}$

$=\frac{5\pi}{6}\mathrm{radians}$ or 2.618 radians

You can use the following chart to convert some of the more common degree measures to radians. You can also use it as a guide to make sure you are in the right range when calculating degrees between the ones in the chart.

Angle in Degrees | Angle in Radians |
---|---|

0° | 0 |

30° | $\frac{\pi}{6}=0.524\mathrm{Rad}$ |

45° | $\frac{\pi}{4}=0.785\mathrm{Rad}$ |

60° | $\frac{\pi}{3}=1.047\mathrm{Rad}$ |

90° | $\frac{\pi}{2}=1.571\mathrm{Rad}$ |

120° | $\frac{2\pi}{3}=2.094\mathrm{Rad}$ |

150° | $\frac{5\pi}{6}=2.618\mathrm{Rad}$ |

180° | $\pi =3.14\mathrm{Rad}$ |

210° | $\frac{7\pi}{6}=3.665\mathrm{Rad}$ |

270° | $\frac{3\pi}{2}=4.713\mathrm{Rad}$ |

360° | $2\pi =6.283\mathrm{Rad}$ |

Let's see how it works for checking our work when we calculate an angle that is not on the chart.

**Example 4**

Convert $40\xb0$ to radian measure.

$40\xb0=40\xb0\times \frac{\pi}{180\xb0}\mathrm{radians}$

$=\frac{\pi}{4.5}\mathrm{radians}$ or (equals about) .698 radians

By looking at the chart, you can see that this answer is between $30\xb0$ and $45\xb0$ , but closer to $45\xb0$ , so it is likely accurate.

Usually, in general geometry, we consider the measure of an angle in degrees. Radian measure is commonly considered when measuring the angles of trigonometric functions or periodic functions. Radian measure is always represented in terms of pi, where the value of pi is equal to $\frac{22}{7}$ or 3.14. If we are working on trigonometric problems, and we measure angles in degrees, it's then necessary to convert the degree measure to radian measure.

Understanding degree and radian measure can be tricky, and converting them from degrees to radians can be downright confusing. If your student is in geometry or trigonometry and is struggling with converting degrees to radians, set them up with an expert tutor who can help them make sense of it all. A private tutor meets with your student in a 1-on-1 setting where they can answer any questions that arise right away so your student understands concepts from the beginning. A tutor will take the time to discover your student's learning style and work at your student's pace, taking extra time when they need it to understand certain concepts. Contact the Educational Directors at Varsity Tutors today to learn more about how tutoring can benefit your student.

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