Lesson 3 of 13 · 7 min
Converting between degrees and radians
NCERT §3.2.4
The fair's poster says the wheel turns 90° every minute. The engineer's manual says π/2 radians per minute. Both are right. How do we move between the two ways of measuring?
The lesson in notes
In short
A full turn is both 360° and 2π radians, so π radians = 180°. Every conversion follows from this one equation.
Radian measure = (π/180) × degree measure, and degree measure = (180/π) × radian measure.
With π = 22/7, 1 radian = 180°/π ≈ 57° 16′, and 1° = π/180 radian ≈ 0.01746 radian.
The common angles to know by heart: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2 and 360° = 2π.
Worked conversions: 75° = 75π/180 = 5π/12 and 150° = 5π/6; going back, 3π/4 = (3/4) × 180° = 135° and 7π/6 = 210°.
A radian value without π is converted the same way. With π = 22/7, 2 radians = 360/π degrees = 2520/22 degrees = 114 6/11°; since 6/11° = 32 8/11′ and 8/11′ ≈ 44″, this is about 114° 32′ 44″.
Convention: θ° means an angle of θ degrees, while a bare letter such as β, or an expression such as π/4, means an angle measured in radians.