Radians and degree conversions

A natural unit for angles

Degrees divide a full turn into 360 parts, a choice inherited from ancient Babylon. Radians measure angles using the circle itself.

1 radian is the angle at the centre of a circle that cuts off an arc equal in length to the radius.

How many radians in a circle?

The circumference is 2πr, so a full turn is 2πr ÷ r = 2π radians.

360° = 2π rad, so 180° = π rad.

Converting

Examples:

• 60° × π/180 = π/3
• 225° × π/180 = 5π/4
• 3π/2 × 180/π = 270°
• 1 rad × 180/π ≈ 57.3°

Common angles

Why use radians?

Formulas become simpler:

• Arc length: s = rθ
• Sector area: A = ½r²θ

Example: r = 10 cm, θ = π/4

• s = 10 × π/4 ≈ 7.85 cm
• A = ½(100)(π/4) ≈ 39.3 cm²

In calculus, the derivative of sin x is cos x only when x is in radians. That’s why radians are the standard unit in higher maths and physics.

Angular speed

Angular speed is how fast something rotates.

Example: A wheel turns at 3 revolutions per second.

• ω = 3 × 2π = 6π rad/s
• If the radius is 0.3 m, the rim’s linear speed is v = rω = 0.3 × 6π ≈ 5.65 m/s

Calculator modes

• DEG mode: sin 90 = 1
• RAD mode: sin(π/2) = 1

Using the wrong mode is a very common source of errors.

Worked examples

Example 1: Convert 150° to radians. 5π/6.

Example 2: Convert 7π/6 to degrees. 210°.

Example 3: Arc length for r = 4 m and θ = 2.5 rad? 10 m.

Common mistakes

• Using degrees in s = rθ.
• Inverting the conversion factor.
• Forgetting to switch calculator modes.

Try it yourself

1. Convert 300° to radians.
2. Convert 5π/3 to degrees.
3. Convert 2 rad to degrees (one decimal).
4. A sector has r = 6 cm and θ = π/3. Find its area.
5. A Ferris wheel turns once every 60 s. Angular speed in rad/s?

Answers: 1) 5π/3 2) 300° 3) 114.6° 4) 6π ≈ 18.8 cm² 5) π/30 ≈ 0.105 rad/s

Quick recap

• A radian is the angle for an arc equal to the radius; 180° = π rad.
• Convert with π/180 or 180/π.
• Radians simplify arc length (s = rθ) and sector area (½r²θ).