Grade 11 trigonometry lesson
Degrees and Radians: Definition, Conversion Formula, Examples, and Practice
Degrees and radians are two units for measuring the same angle turn. A half turn is 180° or π radians.
What are degrees and radians?
Degrees and radians are two ways to measure an angle.
A degree uses 360 equal parts for one full turn. A full turn is 360°.
A radian measures an angle using circle radius and arc length. In trigonometry, radians make circle formulas and graphs easier to use.
The angle can be the same even when the unit changes. For example, 180° and π radians both mean a half turn.
The key conversion fact
Memorize this connection first.
180° = π radians
From that one fact, you can convert between degrees and radians.
Conversion formulas
Use these formulas when the unit needs to change.
Degrees to radians: multiply by π / 180.
Radians to degrees: multiply by 180 / π.
If you later study coterminal angles, these same full-turn values help you add 360° or 2π radians.
Common degree and radian chart
These 15° interval values appear often in trigonometry.
You do not need to memorize the whole chart at once. Start with 0°, 90°, 180°, 270°, and 360°, then add the 15°, 30°, 45°, 60°, and 75° patterns.
Example: convert degrees to radians
Problem: Convert 150° to radians.
Step 1: Multiply by π / 180.
Step 2: 150 × π / 180 = 150π / 180.
Step 3: Simplify the fraction: 150π / 180 = 5π / 6.
Answer: 150° = 5π/6 radians.
Example: convert radians to degrees
Problem: Convert 3π/4 radians to degrees.
Step 1: Multiply by 180 / π.
Step 2: 3π/4 × 180/π = 3 × 180 / 4.
Step 3: 540 / 4 = 135.
Answer: 3π/4 radians = 135°.
Where radians show up
Radians are common in circle-based trigonometry, graphs, and advanced formulas.
A central angle can be measured in degrees or radians because it opens at the center of a circle.
Earlier right-triangle lessons such as angle of elevation often use degrees first, but later trigonometry often switches to radians.
Common mistakes
Do not write π after a degree measure. Use either 180° or π radians, not 180π°.
Do not forget to simplify the fraction when converting degrees to radians.
Do not treat π as the same as 180 in every expression. π radians equals 180°, but π itself is still the number about 3.14.
Quick practice
1. 90° = π/2 radians.
2. 60° = π/3 radians.
3. π/6 radians = 30°.
4. 2π radians = 360°.
Interactive playground
Convert a turn
Move the angle in degrees. The playground shows the matching radian measure.