LEMMA

Trigonometry

Degrees and radians

ALEKS placement

π radians and 180° describe the same half-turn.

What this covers

rad=deg⋅180π​,deg=rad⋅π180​
  • Multiply by π/180 to go from degrees to radians.
  • Multiply by 180/π to go the other way.
  • Leave radian answers exact, in terms of π.
  • Arc length is s = rθ, with θ in radians.

Worked example

Worked example

Convert 135° to radians.

  1. 135⋅180π​Degrees to radians multiplies by π/180.
  2. 43π​135/180 reduces to 3/4.
Another worked example

A circle has radius 6 cm. Find the arc length for a central angle of 120°.

  1. 120⋅180π​=32π​Convert the angle before using the arc-length formula.
  2. s=rθ=6⋅32π​=4π cmMultiply the radius by the angle in radians; this gives a length.

A common mistake

A mistake Lemma catches

From 120⋅180π​=32π​

s=6⋅120s=6⋅32π​

“s = rθ needs θ in radians. 120° is 120 · π/180 = 2π/3, so s = 6 · 2π/3 = 4π ≈ 12.6 cm. Using 120 itself gives 720 cm, more than nineteen times the whole circumference 2π · 6 ≈ 37.7 cm.”

Try one

Sample problem

Convert 120° to radians.

120∘

Practice degrees ↔ radians free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

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