Two Ways to Measure: Radian & Degree Conversions
Degrees and radians are two units for measuring angles. Degrees divide a full rotation into 360 equal parts, while radians relate the angle to the radius of a circle — one full rotation equals 2π radians. Converting between them is essential for trigonometry, calculus, physics, and engineering, where radians are the standard unit.
Components of Radian & Degree Conversions
This section covers the conversion formulas and key reference values:
- Degrees to Radians: Multiply the degree measure by π/180 to convert to radians.
- Radians to Degrees: Multiply the radian measure by 180/π to convert to degrees.
- Common Reference Angles: Memorizing key conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.
- Arc Length with Radians: When the angle is in radians, arc length = r × θ, making radians natural for circular calculations.
Examples of Radian & Degree Conversions
Degrees to Radians Examples
- Convert 90° to radians: 90 × π/180 = π/2 radians.
- Convert 120° to radians: 120 × π/180 = 2π/3 radians.
- Convert 270° to radians: 270 × π/180 = 3π/2 radians.
Radians to Degrees Examples
- Convert π/3 to degrees: (π/3) × 180/π = 60°.
- Convert 5π/6 to degrees: (5π/6) × 180/π = 150°.
- Convert 3π/4 to degrees: (3π/4) × 180/π = 135°.
Common Reference Angles Examples
- 45° = π/4 radians. This is the angle of a right isosceles triangle, where both legs are equal.
- 60° = π/3 radians. In an equilateral triangle, each angle measures exactly π/3 radians.
- 180° = π radians. A straight line represents half a rotation, which is π radians.
Arc Length Examples
- A circle with radius 10 cm has a central angle of π/4 radians. Arc length = 10 × π/4 = 2.5π ≈ 7.85 cm.
- A wheel with radius 0.3 meters rotates through 2π/3 radians. The distance traveled is 0.3 × 2π/3 = 0.2π ≈ 0.63 meters.
- A clock's minute hand is 12 cm long. In 20 minutes (1/3 rotation = 2π/3 radians), the tip travels 12 × 2π/3 = 8π ≈ 25.13 cm.