Angle Conversion Tool
Choose a direction, enter an angle, and control the displayed precision.
Batch Conversion
Convert many angles separated by lines, commas, or semicolons.
Conversion History
Recent results stay in this browser until cleared.
Formula Used
Degrees divide one complete turn into 360 equal parts. Radians measure the same turn using a circle’s radius. One complete turn equals 2π radians.
The conversion factor is π divided by 180. Multiplying degrees by that factor produces radians. Reverse conversion multiplies radians by 180 divided by π.
How to Use This Calculator
- Select the required conversion direction.
- Choose decimal degrees, DMS, decimal radians, or a π multiple.
- Enter the angle and select decimal precision.
- Choose an optional normalization range.
- Select system π or enter a custom π value.
- Press Convert angle to view results and steps.
- Use the copy, share, CSV, PDF, or print controls.
Common Degree and Radian Values
| Degrees | Exact radians | Decimal radians |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.523599 |
| 45° | π/4 | 0.785398 |
| 60° | π/3 | 1.047198 |
| 90° | π/2 | 1.570796 |
| 120° | 2π/3 | 2.094395 |
| 135° | 3π/4 | 2.356194 |
| 180° | π | 3.141593 |
| 270° | 3π/2 | 4.712389 |
| 360° | 2π | 6.283185 |
Worked Conversion Examples
Convert 45 degrees
45 × π ÷ 180 = π/4.
The decimal result is about 0.785398 radians.
Convert 225 degrees
225 × π ÷ 180 = 5π/4.
The decimal result is about 3.926991 radians.
Convert −90 degrees
−90 × π ÷ 180 = −π/2.
The decimal result is about −1.570796 radians.
Understanding Degrees and Radians
A degree is a familiar angular unit used in navigation, construction, and geometry. A full circle contains 360 degrees. A right angle contains exactly 90 degrees.
A radian links an angle directly to circle geometry. One radian subtends an arc equal to the circle’s radius. This relationship makes radians natural for calculus and trigonometry.
Radians also simplify many mathematical formulas. Derivatives of sine and cosine use their cleanest forms with radians. Engineering software often expects angle inputs in radians.
Degrees Versus Radians
| Feature | Degrees | Radians |
|---|---|---|
| Complete turn | 360° | 2π |
| Right angle | 90° | π/2 |
| Common use | Everyday angle measurement | Advanced mathematics and engineering |
| Circle relationship | Divides a turn into 360 parts | Uses arc length divided by radius |
Practical Uses
Programmers convert angles before calling trigonometric functions. Physics calculations use radians for angular velocity and oscillation formulas. Engineers use radians for rotational displacement and control systems.
Surveying and navigation often begin with degree measurements. Robotics software may require those values in radians. The calculator helps move safely between both conventions.
Frequently Asked Questions
How do I convert degrees to radians?
Multiply the degree value by π, then divide by 180.
How many radians are in 180 degrees?
There are exactly π radians in 180 degrees.
How many radians are in 360 degrees?
A complete 360-degree turn equals exactly 2π radians.
Can negative angles be converted?
Yes. Negative degrees produce negative radians unless normalization changes the displayed equivalent angle.
What does angle normalization mean?
Normalization replaces an angle with an equivalent angle inside a selected standard range.
Can I enter degrees, minutes, and seconds?
Yes. Select the DMS input format and enter each component separately.
Why does the exact answer contain π?
π preserves the exact mathematical relationship between degrees and radians.
Can I convert radians back to degrees?
Yes. Swap the conversion direction and enter decimal radians or a multiple of π.
What precision should I select?
Six to eight decimal places usually work well for general calculations.
Does custom π change the exact result?
It changes decimal calculations, while the symbolic π form remains mathematically exact.
Can several angles be converted together?
Yes. Use the batch section and separate values with lines, commas, or semicolons.