Degrees are a human choice; radians come from the circle itself. One radian is the angle you get when you walk one radius along the arc, and from now on most of Pure Mathematics will use it.
One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. A full turn is \(2\pi\) radians, so \(\pi \text{ rad} = 180^\circ\): multiply by \(\dfrac{\pi}{180}\) to go from degrees to radians, and by \(\dfrac{180}{\pi}\) to go back.
Fig. 1: an arc equal in length to the radius r subtends an angle of 1 radian (about 57.3°) at the centre O
Section 01
What is a radian?
Imagine bending a piece of string the length of the radius around the edge of a circle. The angle it makes at the centre is one radian. Because the whole circumference is \(2\pi r\), exactly \(2\pi\) of these radius-lengths fit round the circle, so a full turn is \(2\pi\) radians.
Definition: 1 radian is the angle at the centre subtended by an arc of length equal to the radius.
The key fact: \(2\pi \text{ rad} = 360^\circ\), so \(\pi \text{ rad} = 180^\circ\) and \(1 \text{ rad} = \dfrac{180^\circ}{\pi} \approx 57.3^\circ\).
Notation: an angle with no degree sign is in radians. Write \(\dfrac{\pi}{3}\), or \(1.2\), or \(1.2\) rad, or \(1.2^{c}\); never \(\dfrac{\pi}{3}^\circ\).
Exact answers: where possible give radians as a multiple of \(\pi\), for example \(\dfrac{5\pi}{6}\), unless the question asks for a decimal.
Degrees
30°
45°
60°
90°
120°
135°
150°
180°
270°
360°
Radians
\(\frac{\pi}{6}\)
\(\frac{\pi}{4}\)
\(\frac{\pi}{3}\)
\(\frac{\pi}{2}\)
\(\frac{2\pi}{3}\)
\(\frac{3\pi}{4}\)
\(\frac{5\pi}{6}\)
\(\pi\)
\(\frac{3\pi}{2}\)
\(2\pi\)
Examiner trap
Your calculator must be in the right mode. \(\sin 1.2\) in degree mode gives \(0.0209\) instead of \(0.932\). If a question gives angles in radians (or uses \(\pi\)), switch to radian mode before pressing anything.
Section 02
Converting both ways.
Every conversion is a multiplication by \(\dfrac{\pi}{180}\) or \(\dfrac{180}{\pi}\). A quick check: radians are small numbers (a full turn is only about 6.28), degrees are big ones.
Worked example Degrees to radians
Convert (a) \(225^\circ\) to radians, giving an exact answer, and (b) \(50^\circ\) to radians, giving your answer to 3 significant figures.
(a) Multiply by \(\dfrac{\pi}{180}\): \(225 \times \dfrac{\pi}{180} = \dfrac{225\pi}{180}\).
Cancel by 45: \(\dfrac{225\pi}{180} = \dfrac{5\pi}{4}\).
Do not multiply \(\dfrac{7\pi}{12}\) by 180 and stop: that gives \(105\pi\), not \(105^\circ\). The \(\pi\) in the angle cancels with the \(\pi\) in \(\dfrac{180}{\pi}\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which statement defines one radian?
Quick check+10 XP
Convert \(135^\circ\) to radians.
Quick check+10 XP
Convert \(\dfrac{5\pi}{3}\) radians to degrees.
Quick check+10 XP
In a triangle, two angles are \(\dfrac{\pi}{2}\) and \(\dfrac{\pi}{6}\). What is the third angle?
Section 03
Radians in triangles.
The triangle formulae you already know work unchanged when the angles are in radians, provided the calculator is in radian mode. The only thing that changes is the angle sum: the angles of a triangle add up to \(\pi\), not 180.
Area: \(\tfrac{1}{2}ab\sin C\), with \(C\) the angle between sides \(a\) and \(b\).
Isosceles triangle with apex angle \(\theta\) and equal sides \(r\): the base is \(2r\sin\dfrac{\theta}{2}\), found by splitting it into two right-angled triangles.
Worked example Area and third side
In triangle \(ABC\), \(AB = 8\) cm, \(AC = 5\) cm and angle \(BAC = 1.2\) radians. Find the area of the triangle and the length \(BC\).
The given angle lies between the two given sides, so use \(\tfrac{1}{2}ab\sin C\): area \(= \tfrac{1}{2} \times 8 \times 5 \times \sin 1.2\).
In radian mode, \(\sin 1.2 = 0.93204\), so area \(= 20 \times 0.93204 = 18.64\ldots\)
Answer: \(AB = 6\sqrt{3}\) cm, area \(= 9\sqrt{3}\ \text{cm}^2\)
Your turn
An equilateral triangle has sides of 4 cm. Write each angle in radians and find the exact area.
Each angle is \(\dfrac{\pi}{3}\). Area \(= \tfrac{1}{2} \times 4 \times 4 \times \sin\dfrac{\pi}{3} = 8 \times \dfrac{\sqrt{3}}{2} = 4\sqrt{3}\ \text{cm}^2\).
Examiner trap
Never mix units in an angle sum. If two angles are \(\dfrac{\pi}{4}\) and \(1.1\), the third is \(\pi - \dfrac{\pi}{4} - 1.1\), not \(180 - \dfrac{\pi}{4} - 1.1\).
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to AS & A Level Paper 1 to 6 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
In triangle \(PQR\), \(PQ = 7\) cm, \(PR = 9\) cm and angle \(QPR = 0.8\) radians. Find the area of the triangle.
Exam style+20 XP
Two angles of a triangle are \(\dfrac{2\pi}{5}\) and \(\dfrac{\pi}{4}\). Find the third angle, in terms of \(\pi\).
Exam style+20 XP
Triangle \(OAB\) has \(OA = OB = 10\) cm and \(AB = 12\) cm. Find angle \(AOB\) in radians.
Exam style+20 XP
A triangle has sides 5 cm and 7 cm with an angle of \(\dfrac{2\pi}{3}\) between them. Find the exact length of the third side.
Skill unlocked
Radians, understood.
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