Radians and Degrees: Generalized Angle Measure

A radian is the angle at the center of a circle that subtends an arc equal in length to the radius, and 180° = π radians, so 1 radian ≈ 57.2958° and 1° = π/180 ≈ 0.017453 radians. In brief: to convert degrees to radians multiply by π/180, and to convert radians to degrees multiply by 180/π. For example, 90° = π/2, 60° = π/3 and 2π rad = 360°. In radians the arc length is s = rθ and the sector area is A = (1/2)r²θ.

Radians make formulas in calculus and physics simple: the derivative of sin x is cos x only when x is in radians. A generalized angle measure also extends angles beyond 0° to 360°, to negative angles and to angles larger than a full turn, which is needed for rotation and for periodic functions. This guide explains radians, conversions, arc length and sector area, angles of any size and a conversion table.

What is a radian?

Place the vertex of an angle at the center of a circle with radius r. If the arc between the sides of the angle has length s, the radian measure of the angle is θ = s/r. When s = r, the angle is exactly 1 radian. The full circle has circumference 2πr, so a full turn is 2πr/r = 2π radians, equal to 360°. A half-turn is π radians (180°) and a right angle is π/2.

Radian measure is a ratio of two lengths, so it has no units. The unit “rad” is often written to avoid confusion with degrees. Degrees are a historical division of the circle into 360 parts, a choice commonly traced back to Babylonian astronomy, with the number 360 being convenient because it has many divisors.

How do you convert between degrees and radians?

Use the proportion θ° / 180° = θ_rad / π.

• Degrees to radians: multiply by π/180.
• Radians to degrees: multiply by 180/π.

Example 1

150° = 150 · π/180 = 5π/6 ≈ 2.618 rad. And 45° = π/4 ≈ 0.7854 rad.

Example 2

3π/4 rad = (3π/4) · (180/π) = 135°. And 1 rad = 180/π ≈ 57.30°. And 2 rad ≈ 114.59°.

Example 3: a decimal radian measure

0.75 rad = 0.75 · 57.2958° ≈ 42.97°. Keep the π in exact answers when the angle is a rational multiple of π, such as 7π/12 = 105°.

How do you find arc length and sector area?

With θ in radians: arc length s = rθ and sector area A = (1/2)r²θ = (1/2)rs. In degrees, the formulas are s = (θ/360°) · 2πr and A = (θ/360°) · πr². For the full circle, θ = 2π gives s = 2πr and A = πr².

Example 4

A circle of radius 8 cm and a central angle of 3π/4. Arc length: 8 · 3π/4 = 6π ≈ 18.85 cm. Sector area: (1/2) · 64 · 3π/4 = 24π ≈ 75.40 cm².

Example 5: a Ferris wheel

A Ferris wheel of radius 20 m turns through 1.2 rad. A rider travels 20 · 1.2 = 24 m. In degrees, 1.2 rad ≈ 68.75°.

Example 6: the angle from the arc

An arc of length 15 cm on a circle of radius 6 cm subtends θ = 15/6 = 2.5 rad ≈ 143.2°.

How are angles of any size measured?

In the coordinate plane, an angle is formed by rotating a ray, the terminal side, from a fixed ray, the initial side along the positive x-axis, about the origin. Counterclockwise rotation gives a positive angle and clockwise rotation a negative angle. Angles can be larger than a full turn: 450° = 360° + 90° is one full turn plus a quarter turn.

Two angles with the same terminal side are coterminal. They differ by a whole number of full turns: θ and θ + 360°k (or θ + 2πk). For instance, −60°, 300° and 660° are coterminal. The reference angle of an angle is the acute angle between its terminal side and the x-axis, and it is used to calculate trigonometric values in each quadrant.

Example 7: reducing an angle

To reduce 1,110° to the interval [0°, 360°): 1110 − 3 · 360 = 30°. And −7π/3 + 2π · 2 = 5π/3. The reference angle of 5π/3 (300°) is 60° = π/3.

Degree-radian conversion table

Where do radians come from?

The idea of measuring an angle by the ratio of arc to radius appeared in the work of Roger Cotes in the early eighteenth century, and Leonhard Euler used it systematically in the same period. The term “radian” was first printed in the 1870s, with credit usually given to the engineer James Thomson, the brother of Lord Kelvin, though earlier uses are discussed by historians. The degree measure is much older, linked to Babylonian astronomy and the base-60 number system.

Where are radians used today?

Physics describes rotation with angular velocity in radians per second: a wheel turning at 3 revolutions per second has ω = 6π ≈ 18.85 rad/s. In calculus, the formulas for derivatives and integrals of trigonometric functions are simple only in radians. Programming languages use radians for sin and cos, and the units of angles in engineering are often radians as well.

Which mistakes are common?

Calculator in the wrong mode

sin(π/6) = 0.5 only in radian mode. In degree mode, sin(0.5236°) ≈ 0.009.

Using the formulas s = rθ with degrees

The simple formulas s = rθ and A = (1/2)r²θ require θ in radians.

Using 3.14 for π in exact conversions

Keep π symbolic, as in 5π/6, and use a decimal only at the end.

Mistaking coterminal angles for equal angles

Coterminal angles have the same trigonometric values, but they are different rotations. The measure matters in rotation problems.