Radians and Conversion (AA SL)
Radians are the IB's default angle unit for calculus and circle work, but questions still mix in degrees, so switching cleanly between the two is a skill on its own. This page covers both conversion formulas, when to leave an answer as an exact multiple of \(\pi\) rather than a decimal, and the mode-switching slips that cost marks. It's part of the broader Radians, Arcs & Sectors topic.
18 questions on this sub-topic.
The two conversions
Covered under IB syllabus reference SL3.4, which introduces radian measure of angles alongside arc length and sector area. Neither conversion is in the formula booklet - they're treated as prior knowledge you're expected to know cold.
Degrees to radians
\[\text{radians} = \text{degrees}\times\dfrac{\pi}{180}\]
Multiply by \(\pi/180\); simplify the fraction to leave an exact multiple of \(\pi\).
Radians to degrees
\[\text{degrees} = \text{radians}\times\dfrac{180}{\pi}\]
Multiply by \(180/\pi\); the \(\pi\)s cancel if the radian value is an exact multiple of \(\pi\).
Once an angle is in radians, it feeds straight into arc length and sector area - see Radians, Arcs & Sectors for GDC angle-mode guidance.
Worked examples
\(135^\circ.\)
(a) Convert it to radians (exact).
(b) Convert \(\tfrac{7\pi}{6}\) rad to degrees.
Worked solution
(a) Degrees to radians: multiply by \(\tfrac{\pi}{180}\). M1
\(135\times\tfrac{\pi}{180}=\tfrac{3\pi}{4}\) rad. A1
(b) Radians to degrees: multiply by \(\tfrac{180}{\pi}\).
\(\tfrac{7\pi}{6}\times\tfrac{180}{\pi}=210^\circ.\) A1
Convert to radians (exact).
(a) \(60^\circ\)
(b) \(225^\circ\)
Worked solution
(a) Multiply by \(\tfrac{\pi}{180}\): M1
\(60\times\tfrac{\pi}{180}=\tfrac{\pi}{3}.\) A1
(b) \(225\times\tfrac{\pi}{180}=\tfrac{5\pi}{4}.\) A1
The minute hand of a clock is 9 cm long.
Find the distance its tip travels in 25 minutes, giving your answer in cm.
Worked solution
25 minutes is \(\tfrac{25}{60}\) of a full turn: \(\tfrac{25}{60}\times 2\pi\) M1 \(=\tfrac{5\pi}{6}\) rad. A1
\(\ell=r\theta=9\cdot\tfrac{5\pi}{6}=\tfrac{15\pi}{2}\) M1 \(\approx 23.6\) cm. A1
Convert to radians, in terms of \(\pi\).
(a) \(135^\circ\)
(b) \(210^\circ\)
Worked solution
(a) Multiply by \(\tfrac{\pi}{180}\): M1
\(135\times\tfrac{\pi}{180}=\tfrac{3\pi}{4}.\) A1
(b) \(210\times\tfrac{\pi}{180}=\tfrac{7\pi}{6}.\) A1
A wheel of radius 0.3 m rolls without slipping. Through what angle (in radians) must it turn to travel 9 m, to 3 significant figures?
Worked solution
Distance travelled equals arc length \(s=r\theta\). M1
\(9=0.3\,\theta\Rightarrow \theta=\tfrac{9}{0.3}=30\text{ rad}.\) A1
Common mistakes
- Using \(\theta\) in degrees inside \(s=r\theta\) or \(A=\tfrac12r^2\theta\). Both formulas only work when the angle is in radians - if the question gives degrees, convert first.
- Calculator left in the wrong angle mode. \(\sin1.2\) means something completely different depending on whether the calculator thinks that's \(1.2\) radians or \(1.2^\circ\) - check the mode before evaluating.
- Not simplifying the \(\pi\) fraction. \(135\times\tfrac{\pi}{180}\) left unsimplified as \(\tfrac{135\pi}{180}\) is technically correct but usually loses the accuracy mark - always cancel to lowest terms, e.g. \(\tfrac{3\pi}{4}\).
Ready to practise properly?
19 radian-conversion questions, marked instantly like the real exam.
Quick answers
How do I convert degrees to radians?
Multiply the number of degrees by \(\tfrac{\pi}{180}\). Simplify the resulting fraction to leave an exact multiple of \(\pi\) where possible.
How do I convert radians to degrees?
Multiply the radian value by \(\tfrac{180}{\pi}\). If the radian value is already an exact multiple of \(\pi\), the \(\pi\) terms cancel and you're left with a whole-number degree value.