Spin a wheel once. In degrees that's $360^\circ$, a Babylonian leftover. In radians it's $2\pi$, the number circles themselves chose. By the end of this lesson you'll see why physicists, engineers, and every calculator on Earth quietly prefer the second one.
Today's hook, A Mars rover doesn't think in degrees. Its onboard computer, every engineer in Pasadena, and every physics equation that gets it there all work in radians. Why would NASA quietly ditch the system you've used your whole life?
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Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
Picture a bike wheel finishing one clean spin. In degrees that's $360^\circ$; in radians it's $2\pi \approx 6.28$. Without using a formula which feels more "real" to you, and which would you trust on a billion-dollar rocket?
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The two moves
Work through the core explanation before applying it.
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The two moves
+5 XP to read
There are only two conversions in this entire lesson, and they both spin out of one identity. Lock $\pi$ rad $= 180^\circ$ into muscle memory and the rest is just rearranging.
Every angle in this entire module just travels along one of two roads: multiply by $\frac{\pi}{180}$ to head into radian-land, or multiply by $\frac{180}{\pi}$ to come back out.
$\pi \text{ rad} = 180^\circ$
Degrees → radians
Multiply by $\dfrac{\pi}{180}$. Answer should shrink (you're multiplying by less than 1).
Radians → degrees
Multiply by $\dfrac{180}{\pi}$. Answer should grow that's your sanity check.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
The definition of a radian
How to convert between degrees & radians
Common angle equivalences
Understand
Concepts
Why radians are the natural unit for circular measure
How coterminal angles work in radians
The link between angle, arc length, and radius
Can do
Skills
Convert any angle between systems
Identify quadrants and reference angles in radians
Find positive & negative coterminal equivalents
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Key terms
RadianA unit of angle where one radian subtends an arc equal to the radius.
DegreeA unit of angle where a full rotation is 360°.
Coterminal angleAngles that share the same terminal side; differ by multiples of $2\pi$.
Arc length$l = r\theta$ when $\theta$ is in radians.
Sector area$A = \tfrac{1}{2}r^2\theta$ when $\theta$ is in radians.
Reference angleThe acute angle between the terminal side and the x-axis.
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What is a radian?
Work through the core explanation before applying it.
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What is a radian?
core concept
Forget degrees for a second. Imagine taking the radius of a circle and bending it around the edge like a piece of string. The angle that string carves out at the centre? That's exactly one radian. No formulas, just geometry.
A radian is defined by arc length = radius. One radian ≈ 57.3°.
Because arc length and radius are measured in the same units, radians are technically dimensionless. That's why $\frac{d}{dx}\sin x = \cos x$ only works when $x$ is in radians, it's the natural unit for calculus and physics.
Why NASA uses radians. When calculating spacecraft trajectories, engineers use radians because velocity, acceleration and angular momentum formulas all simplify in the natural unit. Using degrees would inject factors of $\frac{\pi}{180}$ into every calculation, and at billion-dollar stakes, that's a class of error worth removing entirely.
A radian is defined as the angle where the arc length equals the radius: $\theta = \frac{l}{r}$; One radian $\approx 57.3^\circ$; a full revolution = $2\pi$ radians
Pause, copy the radian definition ($\theta = l/r$ when arc length equals radius), the conversion ($1 \text{ rad} \approx 57.3°$), and the full-revolution equivalence ($2\pi \text{ rad} = 360°$) into your book.
Did you get this? True or false: one radian is defined as the angle subtended when the arc length equals the radius of the circle.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · DEGREES → RADIANS
Convert $135^\circ$ to radians, leaving your answer in terms of $\pi$.
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$135^\circ \times \dfrac{\pi}{180^\circ}$
Multiply by $\frac{\pi}{180}$ to convert from degrees.
Find a positive and a negative angle coterminal with $\dfrac{7\pi}{4}$.
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$\dfrac{7\pi}{4} + 2\pi = \dfrac{15\pi}{4}$
Add $2\pi$ for the positive coterminal.
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$\dfrac{7\pi}{4} - 2\pi = -\dfrac{\pi}{4}$
Subtract $2\pi$ for the negative coterminal.
Quick check: Which of these is the correct conversion of $60^\circ$ to radians?
Common errors · the 3 traps that cost marks
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Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
Trap 01
The "leave it ugly" trap
Writing $\frac{120\pi}{180}$ and stopping there will cost you a mark. Markers expect simplified exact values. Cancel common factors before the final line.
Trap 02
Multiplying the wrong way
Some students flip $\frac{\pi}{180}$ and $\frac{180}{\pi}$. Quick gut-check: degrees → radians should shrink the number (you're multiplying by something less than 1).
Trap 03
The half-spin mistake
One full revolution is $2\pi$, not $\pi$. Add only $\pi$ for coterminal angles and you've flipped to the opposite direction. Always work in $2\pi k$.
Did you get this? True or false: to find a coterminal angle you add or subtract multiples of $\pi$.
Quick-fire practice · 5 conversions
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Drill it, then lock it in
Run the quick drill and copy the summary into your book.
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$60^\circ$ to radians
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$\dfrac{3\pi}{2}$ to degrees
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$225^\circ$ to radians
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$-\dfrac{\pi}{6}$ to degrees
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$540^\circ$ to radians
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Revisit your thinking
Earlier you were asked: why might mathematicians and physicists prefer radians over degrees? Radians emerge from the geometry of a circle itself ($\theta = \frac{l}{r}$). They're dimensionless and they make calculus identities like $\frac{d}{dx}\sin x = \cos x$ work without conversion factors. Degrees are a human convention; radians are a mathematical necessity.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 43 marks
Q1. Convert each angle to radians, leaving answers in terms of $\pi$: (a) $240^\circ$ (b) $-135^\circ$ (c) $720^\circ$. Show working. (3 marks)
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ApplyBand 43 marks
Q2. A wheel rotates through $1500^\circ$. (a) Express this in radians. (b) If the wheel has radius 30 cm, how far does a point on the rim travel? (3 marks)
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AnalyseBand 53 marks
Q3. Explain why the radian measure is defined as $\theta = \dfrac{l}{r}$, and use the definition to argue why $\pi$ radians must equal $180^\circ$. (3 marks)
Q3 (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].
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Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
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Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
