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Trigonometry

Degrees and radians

Converts angles between degrees and radians: 180° is π radians.

radians=degrees×π180
Radians to degreesdegrees=radians×180π
One radian1 rad≈57.2958°

Calculator

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See it move

−1.5−1−0.50.511.5−1−0.50.51(cos θ, sin θ)cossin
  • θ = 135° = 3π/4 rad (quadrant II)
  • sin θ = 0.7071, cos θ = −0.7071, tan θ = −1
  • sin²θ + cos²θ = 0.5 + 0.5 = 1

What the symbols mean

r
the angle in radians (rad)
d
the angle in degrees (degrees)
π
pi, about 3.14159

When to use it

Use it whenever a formula expects radians: arc length s=rθ, sector area, angular speed, and all of calculus (the derivative of sin x is cos x only in radians). Degrees are more familiar for everyday angles and triangles. Common pairs worth knowing: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2.

Worked example

Convert 135° to radians.

  1. Start from the formula.

    r=d⁢π180

  2. Substitute d = 135 and work it out.

    r=135⁢π180
    ≈2.35619rad

  3. As an exact multiple of π:

    r=3π4

Answer 135° ≈ 2.3562 radians (three quarters of π)

Why it works

A radian is the angle that cuts off an arc as long as the radius. A full circle's circumference is 2πr, so a full turn is 2π radians, which is 360°. Dividing both by 2 gives 180° = π.

Common mistakes

  • Multiplying by 180/π when converting to radians. Degrees to radians multiplies by π/180.
  • Leaving the calculator in the wrong mode after converting.
  • Thinking π equals 180. π radians is an angle equal to 180°; π itself is 3.14159….