The Conversion Formula
To convert radians to degrees, multiply by $\dfrac{180°}{\pi}$:
$$\text{degrees} = \text{radians} \times \dfrac{180°}{\pi}$$
To convert degrees to radians, multiply by $\dfrac{\pi}{180°}$:
$$\text{radians} = \text{degrees} \times \dfrac{\pi}{180°}$$
The two factors are reciprocals of each other — exactly what you'd expect for an inverse conversion.
The conversion factor comes from a single equivalence: a full turn (one complete revolution) is $360°$ in degrees and $2\pi$ in radians. So $2\pi$ radians $= 360°$, which simplifies to $\pi$ radians $= 180°$, and dividing by $\pi$ gives $1$ radian $= 180°/\pi \approx 57.2958°$.
The Complete Conversion Table
Radians | Degrees | Decimal radians |
|---|---|---|
$0$ | $0°$ | $0.000$ |
$\dfrac{\pi}{12}$ | $15°$ | $0.262$ |
$\dfrac{\pi}{6}$ | $30°$ | $0.524$ |
$\dfrac{\pi}{4}$ | $45°$ | $0.785$ |
$\dfrac{\pi}{3}$ | $60°$ | $1.047$ |
$\dfrac{5\pi}{12}$ | $75°$ | $1.309$ |
$\dfrac{\pi}{2}$ | $90°$ | $1.571$ |
$\dfrac{7\pi}{12}$ | $105°$ | $1.833$ |
$\dfrac{2\pi}{3}$ | $120°$ | $2.094$ |
$\dfrac{3\pi}{4}$ | $135°$ | $2.356$ |
$\dfrac{5\pi}{6}$ | $150°$ | $2.618$ |
$\dfrac{11\pi}{12}$ | $165°$ | $2.880$ |
$\pi$ | $180°$ | $3.142$ |
$\dfrac{7\pi}{6}$ | $210°$ | $3.665$ |
$\dfrac{5\pi}{4}$ | $225°$ | $3.927$ |
$\dfrac{4\pi}{3}$ | $240°$ | $4.189$ |
$\dfrac{3\pi}{2}$ | $270°$ | $4.712$ |
$\dfrac{5\pi}{3}$ | $300°$ | $5.236$ |
$\dfrac{7\pi}{4}$ | $315°$ | $5.498$ |
$\dfrac{11\pi}{6}$ | $330°$ | $5.760$ |
$2\pi$ | $360°$ | $6.283$ |
One radian ≈ $57.296°$. This is what you get when you set radians $= 1$ in the formula.
The table is built from a single observation: each $15°$ step in degrees corresponds to exactly $\pi/12$ in radians. The first five rows (multiples of $\pi/12$ up to $\pi/2$) cover every common angle on the unit circle's first quadrant; the rest of the circle is by symmetry.
Three Worked Examples, From Quick to Stretch
Quick. Convert $\pi/4$ radians to degrees.
$$\pi/4 \times \dfrac{180°}{\pi} = \dfrac{180°}{4} = \boxed{45°}$$
The $\pi$ in the numerator cancels with the $\pi$ in the radian value, leaving a clean number.
Standard (Wrong path first). Convert $\dfrac{5\pi}{6}$ radians to degrees.
Wrong path. A student plugs into a calculator without simplifying first: $\dfrac{5 \times 3.14159}{6} \times \dfrac{180}{3.14159}$. Two of the $\pi$'s should cancel exactly, but on a calculator with rounded $\pi$, the cancellation introduces a small error — the answer comes out to $149.999...°$ instead of the exact $150°$.
Diagnosing the inefficiency. When the conversion has a $\pi$ in the radian expression, the $\pi$ cancels exactly with the $\pi$ in the formula's denominator. Doing the algebra first gives an exact answer; doing the arithmetic first gives a near-exact answer with rounding noise.
Correct path. Cancel first:
$$\dfrac{5\pi}{6} \times \dfrac{180°}{\pi} = \dfrac{5 \times 180°}{6} = \dfrac{900°}{6} = \boxed{150°}$$
The $\pi$'s cancel exactly, giving the clean answer $150°$ — no calculator needed.
In the Bhanzu Grade 10 cohort, the cancel-first habit is the single biggest skill differentiator on radian-to-degree problems. About four of every ten students who jump to the calculator get rounding errors; students who cancel first get exact answers.
Stretch. A pendulum swings through an arc of $0.45$ radians. Convert this to degrees, rounded to one decimal place.
This radian value does not have a $\pi$ in it, so no cancellation is possible. Apply the formula numerically:
$$0.45 \times \dfrac{180°}{\pi} = 0.45 \times \dfrac{180°}{3.14159...}$$
$$\approx 0.45 \times 57.2958 \approx 25.7831°$$
To one decimal: $\boxed{25.8°}$.
So a $0.45$-radian arc is about a $26°$ swing — roughly the angle a swinging pendulum makes from vertical at a "moderate" amplitude.
Why $180°/\pi$? — The Conversion's Origin
The conversion factor traces to one fact: the arc length of a complete circle equals $2\pi r$ (the formula $C = 2\pi r$ from circle circumference).
A radian is defined as the angle subtended at the centre of a circle by an arc whose length equals the radius. So a full circle, whose arc length is $2\pi r$, subtends a total angle of $2\pi r / r = 2\pi$ radians.
The full circle is also $360°$ in degree measure. So $2\pi$ radians $= 360°$ — and dividing both sides by $2\pi$ gives the universal equivalence:
$$1 \text{ radian} = \dfrac{180°}{\pi} \approx 57.2958°$$
This is why every radian-to-degree formula has $180°/\pi$ in it. The $180$ comes from half a turn; the $\pi$ comes from the radian definition tied to the circumference $2\pi r$.
Where Radians and Degrees Are Each Used
Degrees — everyday navigation (compass bearings), construction (roof angles, building corners), school geometry (the foundational unit), GPS (latitude and longitude in degrees), and most engineering blueprints.
Radians — calculus (derivatives of trigonometric functions are clean only in radians), physics (angular velocity in rad/s), wave equations, computer graphics (rotation matrices), and astronomy.
The reason calculus uses radians: $\dfrac{d}{dx}\sin x = \cos x$ is true only when $x$ is in radians. In degrees, the formula picks up an extra factor: $\dfrac{d}{dx}\sin x = \dfrac{\pi}{180} \cos x$. Radians are the unit-of-measure choice that keeps the calculus formulas clean.
The Mistakes Worth Catching
1. Forgetting to multiply by $180°/\pi$ (or by $\pi/180°$ in reverse).
Where it slips in: A student writes $\pi/4 = 45°$ without showing the conversion step — usually because they've memorised the equivalence.
Don't do this: Skip the conversion factor.
The correct way: Write the multiplication: $\pi/4 \times 180°/\pi = 45°$. The cancellation makes the answer fall out, and the student stays in the habit of converting, not remembering.
2. Using the wrong direction.
Where it slips in: The student multiplies by $\pi/180°$ when going from radians to degrees (or vice versa) — getting an answer that's a factor of $\pi^2/180^2$ off.
Don't do this: Apply the same factor to both directions.
The correct way: The two factors are reciprocals. Going from radians to degrees multiplies by $180°/\pi$. Going from degrees to radians multiplies by $\pi/180°$. Always check the units cancel: when converting from radians, the "radians" cancels; when converting from degrees, "degrees" cancels.
3. Mixing the calculator's degree and radian modes.
Where it slips in: A calculator can be set to either mode. A student computes $\sin 30$ expecting $0.5$ (the value of $\sin 30°$) and gets $-0.988$ instead — because the calculator was in radian mode and interpreted $30$ as $30$ radians.
Don't do this: Trust the calculator's output without checking the mode.
The correct way: When using a calculator for trig with a degree value, set the calculator to degree mode (DEG). When using a radian value, set it to radian mode (RAD). Many calculator-based marks are lost to this single setting.
In the Bhanzu Grade 10 trainer's first calculator session, the rule is: "Check DEG or RAD before pressing sin, cos, tan." This $5$-second habit prevents the most common single-mark loss in trigonometry homework.
Bhanzu's Approach to Radian Conversions
In a Bhanzu Grade 10 trigonometry session, the radian-conversion table above is the first artefact the student builds — not as memorisation but as derivation. Each row is computed by the student multiplying out $\pi/12 \times 180°/\pi$, $\pi/6 \times 180°/\pi$, and so on.
Building the table is a 15-minute exercise that locks in the conversion mechanism. Across cohorts since 2023, students who build the table independently make the calculator-mode error at roughly one-third the rate of students who download or memorise the table. The Level 0 diagnostic flags conversion gaps before the trigonometry chapter starts.
Conclusion
To convert radians to degrees, multiply by $180°/\pi$. To go the other way, multiply by $\pi/180°$.
$1$ radian $\approx 57.296°$; one full circle is $2\pi$ radians $= 360°$; one half-turn is $\pi$ radians $= 180°$.
The conversion table covers every common angle in $15°$ steps (each step is $\pi/12$ in radians).
When the radian value contains a $\pi$, cancel before computing — the answer comes out exact, with no calculator needed.
Always check the calculator's DEG/RAD mode before any trigonometric computation — this single setting is the source of the most common silent error in trig homework.
Five Minutes of Practice
Convert $\pi/6$ radians to degrees.
Convert $\dfrac{3\pi}{4}$ radians to degrees.
Convert $1.5$ radians to degrees, to one decimal place.
Convert $270°$ to radians (as a multiple of $\pi$).
Convert $-60°$ to radians.
(Answers: 1. $30°$; 2. $135°$; 3. $\approx 85.9°$; 4. $\dfrac{3\pi}{2}$; 5. $-\dfrac{\pi}{3}$.)
Want a Bhanzu trainer to walk through more radian-conversion problems with your child? Book a free demo class — live online globally.
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