Convert Angles Between Degrees and Radians – Online Calculator

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What Are Degrees and Radians?

Angles are measured in two primary units: degrees (°) and radians (rad).

Degrees: A degree is ¹⁄₃₆₀ of a full circle. This system is widely used in everyday contexts (e.g., navigation, construction).

Radians: A radian is the angle subtended at the center of a circle by an arc equal to the radius. One full circle is 2π radians (~6.283 rad). Radians are essential in calculus, physics, and advanced mathematics.

Key Relationship:

• 360° = 2π radians
• 180° = π radians

Why Convert Between Degrees and Radians?

• Math & Science: Calculus, trigonometry, and physics formulas (e.g., angular velocity) require radians.
• Engineering: Rotational systems and signal processing often use radians.
• Programming: Languages like Python and MATLAB default to radians for trigonometric functions.

How to Convert Degrees to Radians

Formula: Radians = Degrees × π / 180

Example: Convert 45° to radians:

45 × π / 180 = π/4 rad ≈ 0.785 rad

Common Conversions:

• 30° = π/6 rad
• 90° = π/2 rad
• 180° = π rad

How to Convert Radians to Degrees

Formula: Degrees = Radians × 180 / π

Example: Convert (3π/4) rad to degrees:

(3π/4) × 180 / π = 135°

Common Conversions:

• π/3 rad = 60°
• π/6 rad = 30°
• 2π rad = 360°

Visualizing the Relationship

Imagine slicing a pizza:

• Degrees: Each slice is labeled from 0° to 360°.
• Radians: Each slice is labeled in terms of π (e.g., π/2, π).

It's the same angle, just different "units"!

Tips for Avoiding Mistakes

• Always include π in degree-to-radian conversions.
• Check that your calculator is in the correct mode (DEG vs. RAD).
• Simplify fractions when possible: π/4 is cleaner than 0.785 rad.

Real-World Applications

• Physics: Angular displacement and motion.
• Computer Graphics: Rotation in 2D/3D rendering.
• Astronomy: Measuring apparent size and motion of celestial objects.

FAQs About Angle Conversions

Q: Why do mathematicians prefer radians?
A: Radians simplify calculus operations, such as derivatives of sine and cosine.

Q: Can I convert negative angles?
A: Yes! For example, -90° = -π/2 radians.

Q: How can I estimate radians without π?
A: Use π ≈ 3.14. So 60° ≈ 1.05 radians.