Examples · Trigonometry

Degrees to Radians Examples: 10 Worked Conversions

Ten conversions, and one arc length at the end to show what the exact form is for.

6 min read Updated 2026-08-23 Checked by Aziza Smailovic

The standard angles

The first four degrees to radians examples are the ones worth knowing by heart, because they appear in almost every trigonometry question.

1. 30°worked example

30 · π⁄180multiply by the conversion factor
π⁄630/180 reduces to 1/6

2. 45°worked example

45 · π⁄180 = π⁄445/180 is 1/4

3. 60°worked example

60 · π⁄180 = π⁄360/180 is 1/3

4. 90°worked example

90 · π⁄180 = π⁄2a quarter turn

The pattern is worth seeing: the answer is always π over whatever 180 divides by. For 30° that is 6, for 45° it is 4.

Angles that are not standard

The method does not change; only the fraction is less familiar.

5. 135°worked example

135 · π⁄180the conversion
135π⁄180combine into one fraction
3π⁄4divide top and bottom by 45

6. 210°worked example

210π⁄180combine
7π⁄6divide both by 30

7. 300° — a reflex angleworked example

300π⁄180combine
5π⁄3divide both by 60

Negative angles and going back the other way

A negative angle simply turns the other way, and the arithmetic is unchanged.

8. −120°worked example

−120π⁄180 = −2π⁄3the sign travels through

9. 5π/6 back to degreesworked example

5π⁄6 · 180⁄πmultiply by the inverse factor
5 · 180 ÷ 6the π symbols cancel
150°evaluate

Going back is the same operation with the factor inverted. When the angle already contains π, the symbols cancel and the arithmetic becomes easy.

What the exact form is for

The last example shows why 3π/4 is worth keeping rather than writing 2.356.

10. Arc length on a circle of radius 12 cm, angle 60°worked example

60° = π⁄3convert first
arc = rθthe formula, which needs radians
12 · π⁄3 = 4πthe threes cancel
≈ 12.57 cmonly round at the very end

Had the angle been carried as 1.047 radians, the answer would have arrived as 12.566 with no sign that it is exactly 4π. The exact form keeps the structure visible.

Questions about degrees to radians examples

What is 135 degrees in radians?

3π/4. Multiply by π/180 to get 135π/180, then divide top and bottom by 45.

Do I always have to simplify the fraction?

Yes. 135π/180 is correct but unrecognisable; 3π/4 is immediately identifiable as a standard angle.

How do I convert a negative angle?

Exactly the same way. The minus sign travels through the multiplication unchanged.

Why keep π rather than using the decimal?

Because it is exact and it keeps the structure visible. An arc length of 4π says more than 12.566, and rounding early compounds into later steps.