The basic conversion: multiply degrees by π/180
To change degrees to radians, multiply the number of degrees by π and divide by 180. That's the whole formula: radians = degrees × (π/180). If you have 90 degrees, you multiply 90 by π, then divide by 180, which gives you π/2 radians. If you have 180 degrees, you get π radians. If you have 360 degrees, you get 2π radians.
The reason this works is that a full circle is 360 degrees and also 2π radians. So the conversion factor between them is always π/180. Once you know this one relationship, you can convert any angle.
You don't need a calculator for common angles. The ones you'll see most often—30°, 45°, 60°, 90°, 180°, 270°, 360°—convert to clean fractions of π that you can memorize or look up in a table. For unusual angles like 37° or 142°, you'll multiply by π/180 and leave π in your answer unless you need a decimal approximation.
Key Takeaways
- Multiply degrees by π/180 to get radians; this works because 360° equals 2π radians.
- Common angles like 90°, 180°, and 360° convert to simple multiples of π (π/2, π, and 2π).
- You can leave π in your answer as a fraction—π/4 for 45°—or use 3.14159 to get a decimal if your calculator or assignment requires one.
- Radians measure angles by arc length on a circle, which is why they appear in trigonometry, calculus, and physics formulas.
- The conversion works in reverse: multiply radians by 180/π to get degrees back.
Why radians matter more than degrees in math and science
Degrees are arbitrary—someone decided a circle should be 360 parts, probably because 360 divides evenly by many numbers. Radians, by contrast, come directly from the circle itself. One radian is the angle you get when the arc length equals the radius. This natural relationship is why radians show up everywhere in calculus, physics, and engineering.
When you write a sine or cosine function in calculus, the input is always in radians, not degrees. The derivative of sin(x) is cos(x) only if x is in radians. If you use degrees, the formula breaks and you get a messy constant multiplied in. This is why converting to radians matters: it's not just a unit change, it's the difference between formulas that work and formulas that don't.
In physics, angular velocity and angular acceleration are measured in radians per second. Rotational motion, waves, and oscillations all use radians because the math is cleaner and the units make physical sense.
Common angle conversions you should know
Rather than calculating every time, it's faster to remember or recognize these standard conversions:
| Degrees | Radians | Decimal (approximate) |
|---|---|---|
| 30° | π/6 | 0.524 |
| 45° | π/4 | 0.785 |
| 60° | π/3 | 1.047 |
| 90° | π/2 | 1.571 |
| 180° | π | 3.142 |
| 270° | 3π/2 | 4.712 |
| 360° | 2π | 6.283 |
If you're working on a problem and see π/6 or 3π/4, you can immediately recognize these as 30° or 135° without calculating. This recognition saves time and reduces errors.
Step-by-step: converting an unfamiliar angle
Let's say you need to convert 225 degrees to radians. Write out the formula: 225 × (π/180). Simplify the fraction by finding what both 225 and 180 divide by evenly. Both divide by 45, so 225/180 becomes 5/4. Your answer is 5π/4 radians.
Another example: convert 72 degrees. Write 72 × (π/180). Both divide by 36, so 72/180 becomes 2/5. Your answer is 2π/5 radians. If you need a decimal, multiply 2/5 by 3.14159 to get about 1.257.
The key step is always simplifying the fraction before you multiply by π. This keeps your answer clean and makes it easier to spot if you've made an arithmetic error.
Converting radians back to degrees
If someone gives you an angle in radians and you need degrees, use the reverse formula: degrees = radians × (180/π). If you have π/3 radians, multiply by 180/π. The π cancels, leaving you with 180/3 = 60 degrees.
If you have 2.5 radians (a decimal, not a fraction of π), multiply 2.5 by 180/π. That's 2.5 × 180 ÷ 3.14159, which is about 143.2 degrees. The process is the same: multiply by the conversion factor and simplify.
When to use a calculator and when to leave π in your answer
In pure mathematics and most textbooks, leave π in your answer. Write π/4, not 0.785. This is exact and shows you understand the relationship between the angle and the circle. Decimal approximations introduce rounding error and hide the structure of the problem.
In applied fields—engineering, physics labs, programming—you often need a decimal. Use π ≈ 3.14159 or let your calculator handle it. Some programming languages have a built-in constant for π that's more precise than you could type by hand.
If your assignment or textbook specifies which form to use, follow that. Otherwise, the fractional form with π is the safer choice because it's exact.
Frequently Asked Questions
Do I have to memorize the conversion formula?
No. You need to know that 360° = 2π radians, and from that you can derive the formula in seconds. If you forget, remember that a full circle is 360 degrees and 2π radians, so divide both by 360 to get the conversion factor for one degree.
Why is the answer sometimes a fraction with π and sometimes a decimal?
Angles that divide evenly into 360 (like 30°, 45°, 90°) convert to clean fractions of π. Odd angles like 37° or 142° don't simplify nicely, so you either leave them as a fraction times π or use a decimal approximation. Both are correct; it depends on what your problem asks for.
Can I use degrees in calculus, or do I have to switch to radians?
You can use degrees, but the formulas become wrong. The derivative of sin(x°) is not cos(x°)—it's (π/180)cos(x°). Every formula gets an extra constant. It's simpler to convert to radians once and use the standard formulas.
What if I get a negative angle, like −45 degrees?
Use the same formula. −45° × (π/180) = −π/4 radians. Negative angles work the same way; they just point in the opposite direction around the circle.
Is there a shortcut for converting in my head?
For common angles, memorize the table above. For others, remember that π/180 ≈ 0.01745, so you can multiply degrees by roughly 0.0175 to get a decimal radian value. This is fast but approximate; use the exact formula when precision matters.