Degrees to Radians
Angles are an essential concept in mathematics, especially in geometry and trigonometry. They help us understand directions, rotations, and various real-world applications like navigation, engineering, and physics. Two common measurement systems are degrees and radians. Understanding the conversion between these two units is crucial for solving mathematical problems efficiently.
What are Degrees?
Degrees are the most commonly used unit for measuring angles. A full circle is divided into 360 equal parts, and each part represents 1 degree (°).
- 90° – Right Angle
- 180° – Straight Angle
- 360° – Complete Circle
What are Radians?
Radians are another unit for measuring angles, mainly used in higher mathematics and physics. A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
A complete circle measures 2π radians, equivalent to 360°, meaning:
- π radians = 180°
- 1 radian ≈ 57.296°
Degrees to Radians Conversion Formula
The formula to convert degrees into radians:
Radians = Degrees × (π/180)
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Example Conversions:
- Convert 90° to Radians: 90° × (π/180) = π/2 radians
- Convert 120° to Radians: 120° × (π/180) = 2π/3 radians
- Convert 45° to Radians: 45° × (π/180) = π/4 radians
Radians to Degrees Conversion
To convert radians into degrees, use:
Degrees = Radians × (180/π)
Example Conversions:
- Convert π/6 radians to Degrees: (π/6) × (180/π) = 30°
- Convert 2π/3 radians to Degrees: (2π/3) × (180/π) = 120°
Degrees to Radians Chart
| Degrees (°) | Radians |
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | 2π/3 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Applications of Degrees and Radians
- Trigonometry and Geometry: Trigonometric functions like sine, cosine, and tangent use radians for accurate calculations.
- Physics and Engineering: Angular velocity and rotational dynamics are expressed in radians.
- Astronomy and Navigation: Planetary motions and navigation calculations use radians and degrees interchangeably.
- Computer Graphics and Animation: Simulations, graphics rotations, and 3D rendering use radian measures.
Differences Between Degrees and Radians
| Feature | Degrees | Radians |
| Definition | Division of a circle into 360 parts | Arc length equal to radius |
| Full Circle | 360° | 2π radians |
| Usage | Daily life, school math | Advanced math, physics, engineering |
| Trigonometric Calculations | Less efficient | More efficient |
| Symbol | ° (degree) | rad (radian) |
How to Memorize Degrees to Radians Conversion?
- Know that π radians = 180° — use this as the base.
- Memorize common conversions like 30°, 45°, 60°, 90°.
- Practice frequently using tables and quizzes.
- Use mental shortcuts (1 radian ≈ 57.3°).
| Also Check |
| Associative Property |
| Centroid of a Triangle |
| Collinear Points |
| Commutative Property |
| Cos 0 |
| Hypotenuse |
Solved Examples
Q. Convert the following degree measure to radian measure.
a) 28 degrees
b) 20 degrees
Solution: To convert the following degree measure to radians we will use the following steps:
a) 28° × (π/180°) = 7π/45 Radians
b) 20° × (π/180°) = π/9 Radians
Q. Convert 450 degrees into radians.
Solution: By the formula, we know;
Angle in radians = Angle in degree × π/180
Thus,
450 degrees in radians = 450 × π/180 = 7.854 Rad
Q. Convert 18°30’42’’ into the degree.
Solution:
18°30’42’’ = 18° + (30/60)° + (42/(60 x 60))°
= 18° + 0.5° + 0.01166°
= 18.51167°
Q. Convert 200 degrees into radians.
Solution: By the formula, we know;
Angle in radians = Angle in degree × π/180
Thus,
200 degrees in radians = 200 × π/180 = 10π/9 = 3.491 Rad
Degrees to Radians FAQs
How do you convert degrees to radians?
To convert degrees to radians, multiply the degree value by π/180. For example, 90° × (π/180) = π/2 radians.
Is 1 radian equal to 180 degree?
No, 1 radian is not equal to 180 degrees. Instead, π radians = 180 degrees, so 1 radian equals approximately 57.296 degrees.
What is the formula for radians to degrees?
The formula to convert radians to degrees is: Degrees = Radians × (180/π). Simply multiply the radian value by 180 and divide by π.
How do you convert 225 degrees to radians?
To convert 225 degrees to radians: 225 × (π/180) = 5π/4 radians. So, 225 degrees is equal to 5π/4 radians.
What is π in degrees?
In degrees, π radians is equal to 180 degrees.
How to convert degree to radian class 11?
In Class 11 math, you convert degrees to radians using: Radian = Degree × (π/180).
How many radians is 330?
To convert 330 degrees to radians: 330 × (π/180) = 11π/6 radians. Thus, 330 degrees equals 11π/6 radians.
What formula converts radians to degrees?
The formula to convert radians to degrees is: Degrees = Radians × (180/π). Multiply the radian value by 180 and divide by π.
What is the formula for 1 degree to rad?
The formula for 1 degree to radian is: 1° = π/180 radians. You multiply any degree measure by π/180 to find its radian value.