Degrees and radians

Learning outcome

Convert between degrees and radians exactly, distinguish an angle from its numerical measure, and use radian measure for arc lengths and sector areas.

Concepts and assumptions

A degree is 1/360 of a full turn. A radian measures a central angle by the ratio of its intercepted arc length s to the circle’s radius r: θ = s/r, with r > 0 and matching length units.

An arc equal in length to the radius therefore subtends 1 radian. The ratio of lengths has no physical dimension, but writing “rad” prevents confusion with degrees. The same angle has different numerical measures in the two systems.

A full circle has circumference 2πr, so a full turn is 2πr/r = 2π radians = 360°. Halving gives π radians = 180°. Consequently: radian measure = degree measure × π/180; degree measure = radian measure × 180/π.

The denominators 180 and π are nonzero. The constant π is exact; substituting 22/7 or a short decimal would introduce approximation. One radian equals 180/π degrees, approximately 57.30° when rounded to two decimal places.

General rotation angles may be real: positive rotation is anticlockwise and negative rotation clockwise. For the geometric arcs and sectors here, restrict θ to 0 ≤ θ ≤ 2π radians and r > 0.

Rearranging θ = s/r gives s = rθ. A sector occupies θ/(2π) of the circle, so its area is [θ/(2π)] × πr² = r²θ/2. These two formulas require θ in radians. Convert degrees first rather than inserting the degree number directly.

Trigonometric calculator modes must match the input unit. Changing units changes neither the angle nor whether its trigonometric ratios are defined.

Worked examples

Example 1 — Degrees to radians. Convert 150° exactly.

Multiply by π/180: 150 × π/180 = (5/6)π = 5π/6 rad. Check: this is 5/6 of a half-turn, matching 150/180.

Example 2 — Radians to degrees. Convert 7π/12 rad exactly.

Multiply by 180/π: (7π/12) × (180/π) = 7 × 15 = 105°. The nonzero π factors cancel; no approximation is required.

Example 3 — An arc and sector. A circle has radius 12 cm. Find the arc length and sector area subtended by a central angle of 75°.

First convert: θ = 75π/180 = 5π/12 rad. Arc length = 12 × 5π/12 = 5π cm. Sector area = 12² × (5π/12)/2 = 30π cm². Both answers are exact, and their units distinguish length from area.

Common mistakes

Treating radians as lengths; using the conversion factor backwards; inserting degrees into rθ; approximating π before cancellation; using the wrong calculator mode.

Practice questions

1. Convert 225° to radians exactly.
2. Convert 11π/18 rad to degrees.
3. Which is greater, 1 rad or 60°? Give the exact positive difference in radians.
4. A circle has radius 9 cm and a sector angle of 2π/3 rad. Find its arc length and area exactly.

Worked solutions

1. 225 × π/180 = 5π/4 rad. This lies between π and 2π, consistent with an angle between 180° and 360°.
2. (11π/18) × (180/π) = 11 × 10 = 110°.
3. 60° = π/3 rad. Since π > 3, π/3 > 1. Therefore 60° is greater, by (π/3 − 1) rad.
4. Arc length = rθ = 9 × 2π/3 = 6π cm. Area = r²θ/2 = 81 × (2π/3)/2 = 27π cm². No rounding is used.