Radians to Degrees Converter
Convert radians to degrees and degrees to radians, with the answer as a multiple of π, in degrees-minutes-seconds and in gradians, plus the steps.
Results
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Steps
| Step | Working |
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Quick Answer
To convert radians to degrees, multiply by 180 ÷ π. One radian is about 57.2958°, and π radians is exactly 180°. To go the other way, multiply degrees by π ÷ 180, so 90° = π/2 radians.
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Table of Contents
- Radians to Degrees Converter
- Quick Answer
- Related Calculators
- How This Radians to Degrees Converter Works
- How to Use the Radians to Degrees Converter
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Radians vs Degrees
- Common Angles
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Radians to Degrees Converter Works
Choose the direction and enter an angle. For radians you can type a plain number or a multiple of π, such as pi/4 or 3pi/2. The converter gives the angle in degrees or radians, as a multiple of π, in degrees, minutes and seconds, and in gradians, and shows the steps.
Quick explanation: a full circle is 360°, which equals 2π radians.
- 1 radian: about 57.2958°.
- π radians: 180°.
- 1°: about 0.0174533 radians.
How to Use the Radians to Degrees Converter
- Choose radians to degrees or degrees to radians.
- Enter the angle. For radians, you can type pi, such as pi/3.
- Press Calculate.
- Read the converted angle, the π form and degrees-minutes-seconds.
- Follow the steps to see the formula used.
Core Formulas
- Radians to degrees: degrees = radians × 180 ÷ π
- Degrees to radians: radians = degrees × π ÷ 180
- Gradians: degrees × 10 ÷ 9
- Minutes and seconds: 1° = 60′ and 1′ = 60″
Key Terms
- Radian: the angle made by an arc equal in length to the radius.
- Degree: 1/360 of a full turn.
- Gradian: 1/400 of a full turn, also called a gon.
- Arcminute (′): 1/60 of a degree.
- Arcsecond (″): 1/60 of an arcminute.
Worked Example
Convert 2.5 radians to degrees.
- 2.5 × 180 ÷ π
- = 143.2394°
- In degrees, minutes, seconds: 143° 14′ 22.02″
Real-Life Examples
- Trigonometry homework: π/6 = 30°, so sin(π/6) = sin 30° = 0.5.
- Programming: most code uses radians, so 45° becomes 0.7854 rad.
- Physics: an angular speed of 2 rad/s is about 114.6°/s.
- Navigation: convert decimal degrees to degrees, minutes and seconds.
- Engineering drawings: turn a part’s rotation from software in radians into degrees that are easier to measure with a protractor.
Radians vs Degrees
- Degrees: a circle is split into 360 parts. Common in everyday life and navigation.
- Radians: based on the radius. An arc as long as the radius makes 1 radian. Used in calculus and physics.
- Why radians: formulas such as arc length = r × θ are simplest in radians.
Common Angles
| Degrees | Radians | Decimal radians |
|---|---|---|
| 0° | 0 | 0.0000 |
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 120° | 2π/3 | 2.0944 |
| 135° | 3π/4 | 2.3562 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
Common Mistakes to Avoid
- Using the wrong calculator mode. Check whether it is set to degrees or radians.
- Rounding π too early. Keep π exact until the end.
- Mixing up the formulas. Radians to degrees multiplies by 180 ÷ π.
- Forgetting that 1 radian is large. It is about 57°, not 1°.
- Confusing gradians with degrees. A circle has 400 gradians.
Frequently Asked Questions
Multiply by 180 and divide by π. For example, 2 radians × 180 ÷ π is about 114.6 degrees.
Multiply by π and divide by 180.
About 57.2958°.
π radians, about 3.14159.
90°.
60°.
360°, one full turn.
2π, about 6.28319.
They make formulas in calculus and physics simpler, such as the derivative of sin x being cos x.
Type pi or π, such as pi/4, 3pi/2 or 2*pi.
A way to write degrees using 60 minutes per degree and 60 seconds per minute.
A unit where a right angle is 100 gradians and a full circle is 400.
A full circle is both 360 degrees and 2π radians. Dividing both by 2π shows that 1 radian equals 360 ÷ 2π, which simplifies to 180 ÷ π degrees, or about 57.3 degrees.
Use the same formulas. A negative angle simply turns the other way, clockwise instead of counterclockwise. For example, −π/2 radians is −90°, which points the same way as 270°.
Convert as normal, then subtract 360° (or 2π) until the angle is between 0° and 360° if you need the matching direction. The converter shows this as the angle within one turn.
Add the degrees, the minutes divided by 60, and the seconds divided by 3,600. For example, 30° 15′ 36″ = 30 + 0.25 + 0.01 = 30.26°. Then convert to radians if needed.
Most scientific calculators have both modes, often shown as DEG and RAD. Spreadsheets and programming languages usually expect radians in their trig functions, so convert degrees first.
Yes. Because a radian is defined by the ratio of arc length to radius, the angle is the same no matter how big or small the circle is.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Radian | Definition and conversions. | https://en.wikipedia.org/wiki/Radian |
| Wikipedia, Degree (angle) | Degrees, minutes and seconds. | https://en.wikipedia.org/wiki/Degree_(angle) |