Converting between degrees and radians

0/0
Introduction
Lessons
    0/3
    Examples
    Lessons
    1. Review "unit conversion"
      1. 24 roses = ___________ dozens of roses
      2. 3 dozens of roses = ___________ roses
      3. 65 kg = __________ lbs
      4. 100 lbs = __________ kg
    2. Convert the following angles from degrees to radians
      1. 85°
      2. -213°
      3. 30°
      4. 60°
      5. 45°
      6. 90°
      7. 180°
      8. 270°
      9. 360°
    3. Convert the following angles from radians to degrees
      1. π6\pi \over 6
      2. 5π3-{ {5\pi} \over 3}
      3. 2
      4. -8.14
    Free to Join!
    StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
    • Easily See Your Progress

      We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
    • Make Use of Our Learning Aids

      Last Viewed
      Practice Accuracy
      Suggested Tasks

      Get quick access to the topic you're currently learning.

      See how well your practice sessions are going over time.

      Stay on track with our daily recommendations.

    • Earn Achievements as You Learn

      Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
    • Create and Customize Your Avatar

      Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
    Topic Notes

    What is a radian?

    A radian is a type of unit used for measuring angles. The other unit that you're probably more familiar with when it comes to measuring angles is degrees. While there's more than two units that help depict the measure of an angle, radian and degrees are the two that you'll have to deal with most.

    You're probably wondering why we have to use radians when we already have degrees. That is because degrees are not actually numbers. In order to compute math, we need to work with numbers. A good example that's similar to this concept is using decimals when we have percentages. Although a percentage can be shown with a number followed by a % sign, when we do math, we convert that to a decimal (or fraction).

    In this lesson, we'll learn how to convert radians to degrees, and also the other way around: turning degrees into radians.

    Converting radians to degrees

    To start off, let's first convert radians into units that we're more familiar with: degrees.

    Question

    Convert the following angles from radians to degrees

    1. i) π6\frac{\pi }{6}

    Solution

    Same as converting other units, when converting radians to degrees, we need to know the conversion factor. In a sense, this is the radians to degrees formula that can help you change from one unit to another. In the case of converting radian to degree, the conversion factor is 180π  rad\frac{{180^\circ }}{{\pi \;rad}} .

    Now, we can multiply π6\frac{\pi }{6} with the conversion factor 180π  rad\frac{{180^\circ }}{{\pi \;rad}} . The π\pi and rad cancel out each other, and then only 301\frac{{30^\circ }}{1} is left. So, the final answer is 3030^\circ

    Multiply pi/6 by 180/pi to get the answer of 30 degree
    Convert pi/6 to 30 degree
    1. ii) 5π3 - \frac{{5\pi }}{3}

    Similar to the first part of the question, we multiply 5π3 - \frac{{5\pi }}{3} with the conversion factor 180π  rad\frac{{180^\circ }}{{\pi \;rad}}. Again, the π\pi and rad cancel out each other.  Both The 3 and the 180180^\circ are divided by 3. We have 51601 - \frac{5}{1} \bullet \frac{{60^\circ }}{1} left, and we will then multiply them and get the final answer 300 - 300^\circ

    Convert -5pi/3 to -300 degree
    Convert -5pi/3 to -300 degree

    iii) 2

    We multiply 2 with the conversion factor 180π  rad\frac{{180^\circ }}{{\pi \;rad}}. The rad cancels out each other and leaves 2180π2 \bullet \frac{{180^\circ }}{\pi }. Using the calculator, we are able to get the final answer 114.592114.592^\circ

    convert 2 radian to degree
    Convert 2 radian to degree

    iii) -8.14

    We multiply -8.14 with the conversion factor 180π  rad\frac{{180^\circ }}{{\pi \;rad}}. The rad cancels out each other and leaves us with 8.14180π - 8.14 \bullet \frac{{180^\circ }}{\pi }. Using the calculator, we are able to get the final answer 466.388 - 466.388^\circ

    convert -8.14 radian to degree
    Convert -8.14 radian to degree

    Converting degrees to radians

    Conversely, we can convert degrees to radians. Let's take a look at these sets of questions.

    Question

    Convert the following angles from degrees to radians

    1. i) 8585^\circ

    Solution

    The conversion factor for converting degrees to radian is π  radian180\frac{{\pi \;radian}}{{180^\circ }}

    We can multiply 8585^\circ with π  radian180\frac{{\pi \;radian}}{{180^\circ }}. The degree signs cancel out each out. We have 85π  radian18085 \bullet \frac{{\pi \;radian}}{{180}} left. Multiplying them with a calculator, we get the final answer 1.48 radian.

    Similar questions can be done in the same way as this method we just explored:

    Multiply the degrees with conversion factor π  radian180\frac{{\pi \;radian}}{{180^\circ }} . Use calculator for the multiplication when it is necessary. We will then get the final answer.

    As an online resource, if you wanted to quickly convert between radians and decimals, use this online converter to help you out or double check your work.

    180° = π\pi radians