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)

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

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

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