Angles in degrees and radians

An angle measures a rotation of a ray about its end point, from the initial side to the terminal side. Turning anticlockwise gives a positive angle; turning clockwise gives a negative one. Turning right round brings the ray back to the start: one complete revolution. Nothing stops us turning further, so angles larger than 360° make sense too.

Two units are used:

• Degrees: one revolution is 360°, so 1° is 1/360 of a turn. Smaller parts: 1° = 60′ (minutes) and 1′ = 60″ (seconds). These are parts of an angle, not of time.
• Radians: one radian is the angle at the centre of a circle made by an arc as long as the radius. An arc of 2r makes 2 radians, an arc of 3r makes 3, and so on.

Half a circle has arc length πr, so it makes π radians. That gives the key link: π radians = 180°. When an angle is written with no unit, such as sin 2 or π/3, it is in radians.

• Radians → degrees: multiply by 180/π. 5π/18 = 5 × 180/18 = 50°, and 2 radians = 360/π ≈ 114° 35′.
• Degrees → radians: multiply by π/180. 75° = 75π/180 = 5π/12.
• A quick trick: put 180° in place of π. 7π/12 = 7 × 15° = 105°, and 3π/2 = 270°.

Standard angles to know by heart:

When a conversion leaves a decimal, turn the decimal part into minutes, and then seconds. Example: π/16 radians = 180/16 = 11.25° = 11° + 0.25 × 60′ = 11° 15′.

Arc length: an arc of length l on a circle of radius r makes an angle θ = l/r at the centre, so l = rθ, with θ in radians.

• The minute hand of a clock is 21 cm long. In 20 minutes it turns 120° = 2π/3, so its tip moves l = 21 × 2π/3 = 14π ≈ 44 cm.
• Two circles have equal arcs that make 45° and 30° at their centres. Since l = r₁θ₁ = r₂θ₂, the radii are in the ratio r₁ : r₂ = θ₂ : θ₁ = 30 : 45 = 2 : 3.
• A wheel turning 180 times a minute makes 3 turns a second, that is 3 × 2π = 6π radians per second.