Why do mathematicians love radians? Because they make calculus and physics beautiful. A full rotation is not $360$ arbitrary chunks — it is $2\pi$, the natural constant that appears in every circle, wave, and orbit. In this lesson, you will learn to think in radians and convert fluently between degrees and radians.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
A wheel makes one complete turn. In degrees, we say it has turned $360^\circ$. But in radians, we say it has turned $2\pi$. Which do you think is the more "natural" way to measure angles, and why might mathematicians and physicists prefer one system over the other?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Wrong: π radians is exactly 180°.
Right: π radians and 180° represent the same angle, but π is an irrational number (≈3.14159...). The degree system is arbitrary; radians are the natural unit for calculus.
📚 Core Content
A radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius of the circle.
$$1 \text{ radian} = \frac{\text{arc length}}{\text{radius}} = \frac{l}{r}$$
Because the arc length and radius are measured in the same units, radians are technically dimensionless. This is why they appear so naturally in calculus and physics — when you differentiate $\sin x$, the result is $\cos x$ only when $x$ is measured in radians.
The Babylonians chose $360^\circ$ for a full circle because $360$ has many divisors. It is a convenient human convention. But $\pi$ is not a human convention — it emerges from the geometry of circles themselves. A full rotation is $2\pi$ radians because the circumference of a circle is $2\pi r$, so the ratio of circumference to radius is $2\pi$.
| Degrees | Radians |
|---|---|
| $360^\circ$ | $2\pi$ |
| $180^\circ$ | $\pi$ |
| $90^\circ$ | $\frac{\pi}{2}$ |
| $60^\circ$ | $\frac{\pi}{3}$ |
| $45^\circ$ | $\frac{\pi}{4}$ |
| $30^\circ$ | $\frac{\pi}{6}$ |
The conversion formulas are straightforward applications of the fact that $\pi \text{ rad} = 180^\circ$:
With practice, you should be able to convert the common angles instantly without a formula. For less common angles, write out the multiplication clearly.
Two angles are coterminal if they have the same initial and terminal sides. In radians, you can find a coterminal angle by adding or subtracting multiples of $2\pi$.
$$\theta_{\text{coterminal}} = \theta + 2\pi k \quad \text{where } k \in \mathbb{Z}$$
🧮 Worked Examples
🧪 Activities
1 $60^\circ$ to radians
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Answer in your workbook.
2 $\frac{3\pi}{2}$ to degrees
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3 $225^\circ$ to radians
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4 $-\frac{\pi}{6}$ to degrees
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5 $540^\circ$ to radians
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1 $\frac{4\pi}{3}$
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Answer in your workbook.
2 $-\frac{\pi}{4}$
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Answer in your workbook.
3 $\frac{11\pi}{6}$
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Answer in your workbook.
Earlier you were asked: Why might mathematicians and physicists prefer radians over degrees?
Radians are the natural unit of angle measure because they are defined directly from the geometry of a circle: $\theta = \frac{l}{r}$. This makes them dimensionless and causes them to appear naturally in calculus (where $\frac{d}{dx}\sin x = \cos x$ only works in radians) and physics (where angular velocity, frequency, and wave equations all simplify). Degrees are a human convention ($360$ was chosen by the Babylonians); radians are a mathematical necessity.
Now revisit your initial response. What did you get right? What has changed in your thinking?
Look back at your initial response in your book. Annotate it with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
✍️ Short Answer
8. Convert the following angles to radians, leaving answers in terms of $\pi$: (a) $240^\circ$ (b) $-135^\circ$ (c) $720^\circ$. Show your working. 3 MARKS
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Answer in your workbook.
9. A wheel rotates through an angle of $1500^\circ$. (a) Express this angle in radians. (b) If the wheel has a radius of 30 cm, how far does a point on the rim travel during this rotation? 3 MARKS
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Answer in your workbook.
10. Explain why the radian measure of an angle is defined as $\theta = \frac{l}{r}$, and use this definition to argue why $\pi$ radians must equal $180^\circ$. 3 MARKS
Type your answer below:
Answer in your workbook.
1. $\frac{\pi}{3}$
2. $270^\circ$
3. $\frac{5\pi}{4}$
4. $-30^\circ$
5. $3\pi$
1. Quadrant III. Positive coterminal: $\frac{4\pi}{3} + 2\pi = \frac{10\pi}{3}$. Negative coterminal: $\frac{4\pi}{3} - 2\pi = -\frac{2\pi}{3}$.
2. Quadrant IV (or IV if measured clockwise). Positive coterminal: $-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}$. Negative coterminal: $-\frac{\pi}{4} - 2\pi = -\frac{9\pi}{4}$.
3. Quadrant IV. Positive coterminal: $\frac{11\pi}{6} + 2\pi = \frac{23\pi}{6}$. Negative coterminal: $\frac{11\pi}{6} - 2\pi = -\frac{\pi}{6}$.
1. A — $\pi$ rad $= 180^\circ$.
2. A — $\frac{3\pi}{4} \times \frac{180}{\pi} = 135^\circ$.
3. A — $300 \times \frac{\pi}{180} = \frac{5\pi}{3}$.
4. A — $\frac{\pi}{4} + 2\pi = \frac{9\pi}{4}$.
5. A — Full revolution is $360^\circ = 2\pi$.
Q8 (3 marks): (a) $240 \times \frac{\pi}{180} = \frac{4\pi}{3}$ [1]. (b) $-135 \times \frac{\pi}{180} = -\frac{3\pi}{4}$ [1]. (c) $720 \times \frac{\pi}{180} = 4\pi$ [1].
Q9 (3 marks): (a) $1500 \times \frac{\pi}{180} = \frac{25\pi}{3}$ radians [1]. (b) $l = r\theta = 30 \times \frac{25\pi}{3} = 250\pi \approx 785.4$ cm [2].
Q10 (3 marks): A radian is defined as the angle subtended when arc length equals radius [1]. For a semicircle, arc length $= \pi r$, so the angle in radians is $\frac{\pi r}{r} = \pi$ [1]. A semicircle is also $180^\circ$, therefore $\pi$ radians $= 180^\circ$ [1].
Climb platforms using your knowledge of angle measurement and radian conversion. Pool: lesson 1.
Tick when you've finished all activities and checked your answers.