Angles & Radian Measure
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.
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?
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.
Key facts
- The definition of a radian
- How to convert between degrees & radians
- Common angle equivalences
Concepts
- Why radians are the natural unit for circular measure
- How coterminal angles work in radians
- The link between angle, arc length, and radius
Skills
- Convert any angle between systems
- Identify quadrants and reference angles in radians
- Find positive & negative coterminal equivalents
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.
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
Convert $135^\circ$ to radians, leaving your answer in terms of $\pi$.
Convert $\dfrac{5\pi}{6}$ radians to degrees.
Find a positive and a negative angle coterminal with $\dfrac{7\pi}{4}$.
Quick check: Which of these is the correct conversion of $60^\circ$ to radians?
Common errors · the 3 traps that cost marks
Did you get this? True or false: to find a coterminal angle you add or subtract multiples of $\pi$.
Quick-fire practice · 5 conversions
$60^\circ$ to radians
$\dfrac{3\pi}{2}$ to degrees
$225^\circ$ to radians
$-\dfrac{\pi}{6}$ to degrees
$540^\circ$ to radians
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.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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)
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)
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)
📖 Comprehensive answers (click to reveal)
Drill 1: $\frac{\pi}{3}$ · 2: $270^\circ$ · 3: $\frac{5\pi}{4}$ · 4: $-30^\circ$ · 5: $3\pi$
Q1 (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].
Q2 (3 marks): (a) $1500 \times \frac{\pi}{180} = \frac{25\pi}{3}$ rad [1]. (b) $l = r\theta = 30 \times \frac{25\pi}{3} = 250\pi \approx 785.4$ cm [2].
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].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering radian conversions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review.