How To Convert Radians To Degrees

Most everyone has heard about pi. Pi is the ratio of the circumference of a circle to its diameter. Degrees measure arc length on the x-axis of a graph. But do you know how often degrees and radians are used together? I’ll try to answer that question in my post about How To Convert Radians To Degrees using pi conversions.

Radians and degrees are both units used for measuring angles. As you may know, a circle is comprised of 2π radians, which is the equivalent of 360°; both of these values represent going “once around” a circle. Therefore, 1π radian represents going 180° around a circle, which makes 180/π the perfect conversion tool for moving from radians to degrees. To convert from radians to degrees, you simply have to multiply the radian value by 180/π. If you want to know how to do this, and to understand the concept in the process, read this article.

Steps

  1. 1 Know that π radians is equal to 180 degrees. Before you begin the conversion process, you have to know that π radians = 180°, which is equivalent to going halfway around a circle. This is important because you’ll be using 180/π as a conversion metric. This is because 1 radians is equal to 180/π degrees.[1]
  2. 2 Multiply the radians by 180/π to convert to degrees. It’s that simple. Let’s say you’re working with π/12 radians. Then, you’ve got to multiply it by 180/π and simplify when necessary. Here’s how you do it:[2]
    • π/12 x 180/π =
    • 180π/12π ÷ 12π/12π =
    • 15°
    • π/12 radians = 15°
  3. 3 Practice with a few examples. If you really want to get the hang of it, then try converting from radians to degrees with a few more examples. Here are some other problems you can do:
    • Example 1: 1/3π radians = π/3 x 180/π = 180π/3π ÷ 3π/3π = 60°
    • Example 2: 7/4π radians = 7π/4 x 180/π = 1260π/4π ÷ 4π/4π = 315°
    • Example 3: 1/2π radians = π /2 x 180/π = 180π /2π ÷ 2π/2π = 90°
  4. 4 Remember that there’s a difference between “radians” and “π radians.” If you say 2π radians or 2 radians, you are not using the same terms. As you know, 2π radians is equal to 360 degrees, but if you’re working with 2 radians, then if you want to convert it to degrees, you will have to calculate 2 x 180/π. You will get 360/π, or 114.5°. This is a different answer because, if you’re not working with π radians, the π does not cancel out in the equation and results in a different value.

Most trig applications deal with degrees – in fact, our brains naturally tend to think in terms of degrees too. Haven’t you heard the phrase, “he turned a 180” or “make a 360”?

Degrees just comes naturally to us.

So why change?

Well, the problem with only working with degree measure is that it limits our ability to apply angles to other functions because we’re stuck with values between 0 and 360.

In fact, as Purple Math explains, a degree is not a number we can do most mathematical computations with. It’s very similar to the idea between a percent and a decimal.

If I said, we have used up 50% of our storage space, we all have a clear picture. But, if we wanted to do any mathematical computation then we have to convert it to a useful number, which means we have to convert it to its decimal form of 0.5.

So how do we fix the problem?

Radians! If we convert degrees into radian measure, then we are allowed to treat trigonometric functions as functions with domains of real numbers rather than angles!

What is a radian?

Okay, so radian is an angle with vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle. Or as summarized by Teacher’s Choice, one radian is the angle of an arc created by wrapping the radius of a circle around its circumference.

Huh?

Picture a circle.

Now we know two things:

  • A circle has 360 degrees all the way around.
  • The circumference of any circle is just the distance around it. This means that the number of radii in the circumference is 2pi.

Which means that one trip around a circle is 360 degrees or 2pi radians!

Formula for the Circumference of a Circle

I still don’t get it.

Here’s another way to look at it…

Remember when we created our unit circle? Well, we specified that our radius was the value of 1, right? So, if we then want to calculate our circumference of this unit circle, our distance around would be 2pi.

Ah, now I see. So in our unit circle, we have a circumference of 2pi, which means I’ve gone all the way around which is just like rotating 360 degrees.

Degrees to Radians Formula

Well, now that we know that 360 degrees (rotational measure) equals 2pi radians (distance measure), we can switch back and forth quickly and easily. In fact, we make it even nicer by simplifying and using the conversion: 180 degrees = pi radians!

Degrees to Radians Formula

But why do we even have to do this? This seems like a lot of work, and I’m already happy with degrees – they’re easy and comfortable.

In more advanced mathematics, the use of radian measure is preferred and often required to solve problems. To deal limits and derivatives, which helps us to explain how things change over time, we have to use radians – and over time, you will see that radians are easy, fun, and very, very helpful!

Trust me…

…radians are friends!

Conclusion

There are two common angles unit that are used for measuring the size of any angle, which are Degrees and Radians. Degrees are the most frequently used and known unit of measuring angles and often we just say ‘angle’ and meaning degrees. While radians seem to be forgotten and we should use them more often instead of degrees!

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