Convert radians to degrees
PrintRadian to Degree Conversion Table
| Radian (rad) | Degree (°) |
|---|---|
| 0.01 rad | 0.5729578° |
| 0.1 rad | 5.729578° |
| 0.5 rad | 28.64789° |
| 1 rad | 57.29578° |
| 1.570796 rad (\(\pi / 2\)) | 90.0° |
| 3.141593 rad (\(\pi\)) | 180.0° |
| 6.283185 rad (\(2\pi\)) | 360.0° |
| 10 rad | 572.9578° |
In geometry, trigonometry, and physics, angles are primarily measured in either degrees (°) or radians (rad). While degrees divide a full circular rotation into 360 arbitrary increments, radians provide a natural mathematical relationship based on the circle’s radius: 1 radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
Our free online Radians to Degrees Converter provides instant bidirectional angle conversions:
- Decimal Radians to Degrees: Multiply radians by
180 / π(approx.57.29577951). - Exact π Radians to Degrees: Cancel out
πand simplify the resulting integer or fraction. - Degrees to Radians: Multiply degrees by
π / 180(approx.0.01745329).
Radians to Degrees Formula & Step-by-Step Calculation
The fundamental relationship between radians and degrees is:
Angle in Degrees (°) = Angle in Radians (rad) × (180 / π)
Example 1: Convert 2.5 Radians to Decimal Degrees
| Step | Execution Logic | Calculated Value |
|---|---|---|
| Step 1: Input Value | Starting radian angle | 2.5 rad |
| Step 2: Conversion Multiplier | Multiply by (180 / π) | 2.5 × (180 / 3.14159265) |
| Step 3: Final Answer | 2.5 × 57.2957795 |
143.24° (143° 14′ 22″) |
Example 2: Convert π Fractions to Exact Degrees (With π Cancellation)
When given an angle containing π (such as 3π/4 rad), π cancels out algebraically:
| Radian Input | Substitution Formula | π Cancellation Step | Exact Degrees |
|---|---|---|---|
π/6 rad |
(π/6) × (180/π) |
180 / 6 |
30° |
π/4 rad |
(π/4) × (180/π) |
180 / 4 |
45° |
π/3 rad |
(π/3) × (180/π) |
180 / 3 |
60° |
π/2 rad |
(π/2) × (180/π) |
180 / 2 |
90° (Right Angle) |
3π/4 rad |
(3π/4) × (180/π) |
(3 × 180) / 4 = 540 / 4 |
135° |
π rad |
π × (180/π) |
180 |
180° (Straight Angle) |
3π/2 rad |
(3π/2) × (180/π) |
(3 × 180) / 2 = 540 / 2 |
270° |
2π rad |
2π × (180/π) |
2 × 180 |
360° (Full Circle) |
Unit Circle Trigonometry Lookup Table (0 to 2π Radians)
Below is the standard unit circle reference table containing exact radians, decimal radians, degrees, and quadrant positions:
| Exact π Radians | Decimal Radians | Degrees (°) | Circle Position / Quadrant |
|---|---|---|---|
| 0 rad | 0.0000 rad | 0° | Positive X-Axis (Start) |
| π/6 rad | 0.5236 rad | 30° | Quadrant I |
| π/4 rad | 0.7854 rad | 45° | Quadrant I (45° Bisector) |
| π/3 rad | 1.0472 rad | 60° | Quadrant I |
| π/2 rad | 1.5708 rad | 90° | Positive Y-Axis (Perpendicular) |
| 2π/3 rad | 2.0944 rad | 120° | Quadrant II |
| 3π/4 rad | 2.3562 rad | 135° | Quadrant II |
| 5π/6 rad | 2.6180 rad | 150° | Quadrant II |
| π rad | 3.1416 rad | 180° | Negative X-Axis (Half Circle) |
| 7π/6 rad | 3.6652 rad | 210° | Quadrant III |
| 5π/4 rad | 3.9270 rad | 225° | Quadrant III |
| 4π/3 rad | 4.1888 rad | 240° | Quadrant III |
| 3π/2 rad | 4.7124 rad | 270° | Negative Y-Axis (Three-Quarter Circle) |
| 5π/3 rad | 5.2360 rad | 300° | Quadrant IV |
| 7π/4 rad | 5.4978 rad | 315° | Quadrant IV |
| 11π/6 rad | 5.7596 rad | 330° | Quadrant IV |
| 2π rad | 6.2832 rad | 360° | Full 360° Revolution (Co-terminal with 0°) |
Converting Decimal Degrees to Degrees, Minutes, and Seconds (DMS)
When converting decimal degrees into astronomical and navigational precision:
- The integer portion is the Degrees (°).
- Multiply the decimal remainder by 60 to get Minutes (‘).
- Multiply the remaining decimal fraction by 60 to get Seconds (“).
Example: Convert 1 radian (57.2957795°) into DMS format:
- Degrees: 57°
- Minutes:
0.2957795 × 60 = 17.74677'→ 17′ - Seconds:
0.74677 × 60 = 44.8"→ 45″ - Result:
1 Radian = 57° 17' 45"
Related Trigonometry and Angle Calculators
Need to convert degrees back to radians or calculate trigonometric values? Explore our related math tools:
- Degrees to Radians Converter — Convert angles from degrees to radians and exact π radians.
- Trigonometry Calculator — Calculate sin, cos, tan, sec, csc, and cot values.
- Circle Calculator — Calculate radius, diameter, circumference, and sector area.
- Master Conversion Calculator — Convert units across length, power, mass, speed, and volume.
Frequently Asked Questions (FAQ)
How many degrees are in 1 radian?
There are approximately 57.29578 degrees (or 57° 17' 45") in 1 radian.
What is the easiest way to convert radians with π to degrees?
Replace π with 180° and perform standard division. For example: π/4 = 180° / 4 = 45°; 5π/6 = (5 × 180°) / 6 = 900° / 6 = 150°.
Why are radians preferred over degrees in calculus and physics?
Radians are the natural mathematical unit of angle measurement because they directly link linear arc length to circular radius ($s = r\theta$). In calculus, derivative formulas for trigonometric functions such as $\frac{d}{dx}(\sin x) = \cos x$ only hold true when $x$ is expressed in radians.
How many radians are in a full 360° circle?
There are exactly 2π radians (approximately 6.283185 rad) in a complete 360° circular revolution.