Geometric Angles: Converting Degrees to Radians

Degrees and radians are essential units of your mathematical toolbox, having a degree of significance in all mathematical courses following geometry(did you catch the pun?). 😅💭

Need a quick review on degrees and radians and how to convert between the two for an upcoming exam? Perhaps you stumbled upon this resource and might be curious about the topic. No matter why you’re here, we’re excited to provide you with this quick guide that will guide you through everything you’ll need to know about degrees, radians, and conversions! 🌞✌

Are you ready? Let’s get started! 🚀

Degrees in the Unit Circle 

The unit circle has many uses for mathematicians, including discovering a value in degrees or angle measurements. 📐

Must-Knows 🤔

• There are 360 degrees in the unit circle.

• The unit circle is made up of four equal quadrants that have 90 degree angles (four right angles) each.

Feel free to ignore the trig stuff (sin, cos, tan, cot, sec, csc)!

Image courtesy of Wikipedia

Radians in the Unit Circle

The other helpful value for mathematicians, particularly for the unit circle, is radians, or the measurement of an arc length. How can we identify the length of an arc length? Let’s put some formulas you might know into practice to discover this! 💪

C = 2πr, where C is the value of the circle's circumference, and 2πr is equivalent to pi multiplied by the circle's radius, multiplied by two to account for 2r, which is equal to the diameter of a circle! (C = 2πr is equivalent to C = πd)

Using the formula C = 2πr, we know that the circumference (the arc length) is 2. We can disregard the value r because that defines the distance between the center and the circumference of the circle and is irrelevant.

Discovering the Relationship: Degrees & Radians

If we know that the circumference of a circle is C = 360° = 2π, we could also recognize that 0.5C = 180° = π. Therefore, every semicircle has an angle of 180° and an arc length, or circumference distance of π. Let’s put this into the context of a pie to identify the relationship between degrees and radians, then find the conversion factors! 🤯

Figure not drawn to scale.

In the figure above, let’s visualize this as a pie with a slice being cut from it! We know that this is a quarter of the pie because the angle between the two lines is 90 degrees. We also know that π = 180°. Since we know that 180/2 = 90, 90° must be equivalent to π/2! 🥧

Let’s try another example together! 🆒

Figure not drawn to scale.

This second figure shows a “slice of pie” with an angle of 45 degrees. Knowing that π = 180°, let’s try and find a conversion that will allow us to find some relationship or “conversion factor” that will make it easier to find a way to convert from one unit to the other. ❌

Since we know that π = 180°, we can represent this as πradians/180 degrees because if both are the same value (such as 1/1 or 2/2), they produce the value “1.” Let’s set this equal to the number of radians over the number of degrees (similar to πradians/180 degrees) and simplify! 🧱

Notice that we multiplied the degrees by x radians by 𝜋/180 to make the conversion.

What if you were to try and find the number of degrees if you were given radians? Notice that this new process would inverse the process we’ve detailed above because we would be completing the opposite process! Let’s try another, more complex process, this time utilizing the “inverse” method as an example.

Final Conversions

Great work on discovering the conversion factors! Here’s a quick recap of our discoveries:

Conversion Practice Problems

Think you’ve got a handle of the conversion processes? Let’s give it a try with a couple of problems. You can find the answers for them at the bottom of this page.

Problem 1️⃣:

Convert 𝜋/3 radians to degrees.

Problem 2️⃣:

Convert 300 degrees to radians.

Problem 3️⃣: 

Convert 410 degrees to radians.

Practice Problem Answers

Problem 1 Answer: 60 degrees.

• (𝜋/3)(180/𝜋) =

• 60 degrees

Problem 2 Answer: 5𝜋/3 radians.

• (300 degrees)(𝜋/180) =

• 300𝜋/180 =

• 5𝜋/3 radians

Problem 3 Answer: 41𝜋/18 radians

• (410 degrees)(𝜋/180) =

• 410𝜋/180 =

• 41𝜋/18 radians

What to Take Away

Converting between radians and degrees is fairly simple! Because we know that 180 is equivalent to 𝜋, we can calculate by multiplying the value by 180/𝜋 or 𝜋/180 to convert from one unit to the other. Good luck with your studies! 👌