To convert between degrees and radians, use one fact: 180° = π radians. Everything else is a fraction of that. Conversion appears at the start of almost every arc, sector and trigonometry question in radians, arcs and sectors.
What is a radian?
Take a circle of radius r and cut off an arc that is also of length r. The angle at the centre for that arc is one radian. Since the whole circumference is 2πr, a full turn contains 2π radians, which is 360°.
Halving both sides gives the anchor fact: 180° = π rad. A right angle is π/2, and 60° is π/3.
How do you convert, step by step?
- Degrees to radians: multiply by π/180, then simplify the fraction.
- Radians to degrees: multiply by 180/π.
- If the radian value has π in it, the π cancels and you are left with a whole number of degrees.
- Check the size: a radian is about 57°, so 2 rad should be a little over 100°.
The check in step 4 catches a flipped fraction straight away. If 2 rad came out as 0.035°, something is upside down.
Worked example
Convert (a) 135° to radians, (b) 5π/12 rad to degrees, (c) 2.4 rad to degrees, to 1 decimal place.
(a) 135 × π/180 = 135π/180. Divide top and bottom by 45 to get 3π/4. So 135° = 3π/4 rad.
(b) 5π/12 × 180/π. The π cancels, leaving 5 × 180/12 = 5 × 15 = 75. So 75°.
(c) 2.4 × 180/π = 432/π = 137.50987… So 137.5° to 1 decimal place.
Size check for (c): 2.4 rad is between 2 rad (about 115°) and 3 rad (about 172°), so 137.5° is sensible.
The mistake to watch for
A common slip is to use the calculator in the wrong mode, or to treat a radian number as if it were degrees.
Mistaken working: sin 1.2 = 0.0209
The calculator was in degree mode, so it worked out sin 1.2°. The question meant 1.2 radians.
The correct value is sin 1.2 = 0.9320 to 4 decimal places, with the calculator in radian mode. A quick test: sin 1 should give 0.8415 in radian mode and 0.0175 in degree mode. Run that test at the start of every paper.
Check yourself
1. Convert 72° to radians, in terms of π.
Show answer
72 × π/180 = 72π/180. Divide by 36: 2π/5.
2π/5 rad
2. Convert 7π/9 rad to degrees.
Show answer
7π/9 × 180/π = 7 × 180/9 = 7 × 20 = 140.
140°
3. Convert 0.8 rad to degrees, to 1 decimal place.
Show answer
0.8 × 180/π = 144/π = 45.8366…
45.8°. Size check: 0.8 rad is a little under a right angle, which is about 57° × 0.8.
Where this leads next
With conversion secure, move on to finding an arc length, then test the whole module with the radians, arcs and sectors practice set. The non-calculator working trainer helps you keep answers exact with π.
Some students can convert on a clean page but slip when the angle arrives inside a longer question. That is the kind of pattern our teachers look for in online one-to-one Additional Mathematics tuition.