A full turn is 360° or 2π radians, so π radians equals 180°. To convert, multiply by the fraction that cancels the unit you are leaving.
This lesson opens circular measure. The next lesson, arc length and sector area, uses the radian form.
Which fraction do you use?
| Direction | Multiply by | Why |
|---|---|---|
| Degrees to radians | π ÷ 180 | The degree symbol cancels |
| Radians to degrees | 180 ÷ π | The π cancels |
Do a size check after converting. A radian is about 57°, so 1.8 radians should be a little over 100°, and 72° should be a little over 1 radian.
Worked example: three conversions
Convert 72° to radians. Multiply by π ÷ 180: 72 × π ÷ 180 = 2π/5. As a decimal, 2π/5 = 1.257 radians (to 3 decimal places).
Convert 150° to radians. 150 × π ÷ 180 = 5π/6.
Convert 1.8 radians to degrees. Multiply by 180 ÷ π: 1.8 × 180 ÷ π = 103.1° (to 1 decimal place).
Check the last one by size: 1.8 radians is a little under 2 radians, and 2 radians is about 114.6°, so 103.1° is sensible.
The mistake that costs marks
A common slip is to use the upside-down fraction. Take 72° again.
| Step | Wrong | Right |
|---|---|---|
| Choose the fraction | 180 ÷ π | π ÷ 180 |
| Working | 72 × 180 ÷ π | 72 × π ÷ 180 |
| Result | 4 125.3 | 1.257 |
| Size check | Far above 2π, impossible for 72° | Just over 1, sensible |
An angle of 72° cannot be over 4 000 radians. The habit is to ask whether the answer fits a full turn of 6.28 radians.
Check yourself
Convert 225° to radians in exact form, and 0.75 radians to degrees correct to 1 decimal place.
Answer
225 × π ÷ 180 = 5π/4, which is 5π/4 radians. As a decimal this is 3.927, which is between π and 3π/2, as expected for an angle between 180° and 270°.
0.75 × 180 ÷ π = 42.97, so 43.0° to 1 decimal place.
What to study next
Go on to calculating arc length and sector area, then test yourself with the circular measure practice set.
The word-problem structure worksheet helps you write down what each question gives, and a mistake log shows whether you keep flipping the fraction. To work with a teacher, see online one-to-one Additional Mathematics tuition.