Key takeaways
- radians = degrees × π / 180.
- 180° equals exactly π radians.
- A full circle is 360° or 2π radians.
- Most math libraries and calculus formulas expect radians, not degrees.
In this guide
What is a radian?
One radian is the angle you get when the arc length along a circle equals the circle's radius. A full circle is 360° or 2π radians, so radians and degrees are just two different rulers for measuring the same rotation.
Why does this conversion matter?
Degrees feel natural to most people, but calculus, physics formulas, and programming languages almost all use radians internally. If you feed a formula an angle in degrees when it expects radians, the answer comes out completely wrong, even though nothing throws an error.
A 90° turn looks small in degrees, but it converts to about 1.5708 radians — roughly a quarter of the full 2π ≈ 6.2832 radians in a circle.
The formula
| Symbol | What it means | Example |
|---|---|---|
| degrees | The angle you're starting with | 90° |
| π (pi) | A fixed number, about 3.14159 | 3.14159 |
| radians | The same angle, measured in radians | 1.5708 |
How to convert degrees to radians step by step
- 1Start with your angle in degrees.
- 2Multiply it by π (3.14159).
- 3Divide the result by 180.
- 4The answer is the angle in radians.
Enter any angle in degrees below to see the radian conversion worked out step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting π entirely and just dividing degrees by 180 — that skips a key part of the formula.
- Mixing units mid-calculation. Trig functions in code (and the sin, cos, tan calculator) usually expect radians, so convert before you calculate, not after.
- Rounding π too early. Keep several decimal places until the final answer for anything precision-sensitive.
Tips and tricks
- To go the other direction, use the radians to degrees calculator.
- Radians are also what you need to find the length of a circular arc — see the arc length calculator.
- If your answer looks unreasonably large, you probably multiplied by 180 instead of dividing.
Where you'll use it in real life
- Programming: JavaScript's
Math.sin(), Python'smath.cos(), and most math libraries expect radians. - Physics: angular velocity and rotational motion formulas are built around radians.
- Calculus: derivatives of sine and cosine only simplify cleanly when the angle is in radians.
- Engineering: CAD and robotics software often specify rotations in radians internally.
Quick summary
To convert degrees to radians, multiply by π and divide by 180. Radians are the default angle unit almost everywhere in math and code beyond basic geometry, so this conversion is one you'll use often. Try the calculator above with your own angle.
Worked example
A robotic arm's control software needs angles in radians, but the engineer designed the rotation as 90° on paper. What is 90° in radians?
- 01radians = degrees × π / 180
- 02radians = 90 × π / 180
- 03radians = 0.5π
- 04radians = 1.5708
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I convert degrees to radians?
Multiply the angle in degrees by π, then divide by 180: radians = degrees × π / 180.
What is 180 degrees in radians?
180° equals exactly π radians (about 3.14159), since half a full turn in radians is π.
Why do programming languages use radians instead of degrees?
Radians connect directly to the circle's radius and simplify calculus formulas (like derivatives of sine and cosine), so most math libraries default to them.
What is 1 degree in radians?
1° is about 0.0175 radians. It's a small number because a full circle is only about 6.28 radians but 360 degrees.
