What Is the Degrees to Radians Formula?
The conversion is one line:
$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$
To turn any angle in degrees into radians, multiply it by $\frac{\pi}{180}$. A degree is $\frac{1}{360}$ of a full turn, a unit inherited from Babylonian astronomy. A radian is the angle a circle's arc subtends when that arc is exactly as long as the radius, a unit built from the circle itself rather than an arbitrary count of 360. Because both measure the same angles, a fixed factor converts between them.
Worked quickly, $90^\circ$ becomes:
$$90 \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians}$$
Leaving the answer as a multiple of $\pi$ is standard practice; $\frac{\pi}{2}$ is more exact and more useful than its decimal $1.5708$. Only convert to a decimal when a problem demands a number.
Why Do You Multiply by π/180?
Where does the $\frac{\pi}{180}$ actually come from? From one anchor fact: a straight angle spans 180° and π radians at the same time. This is the single most-asked question about the conversion on forums, and the answer is short once you see the equivalence.
Start from the full circle. A complete turn is $360^\circ$, and in radians it is $2\pi$ (because the full circumference $2\pi r$ wraps $2\pi$ radius-lengths around the centre). Set them equal and halve:
$$360^\circ = 2\pi \text{ radians} \quad\Rightarrow\quad 180^\circ = \pi \text{ radians}$$
Divide both sides by 180 to find what one degree is worth:
$$1^\circ = \frac{\pi}{180} \text{ radians}$$
So multiplying any degree count by $\frac{\pi}{180}$ scales it into radians. The factor is not a magic number; it is $\frac{\pi \text{ radians}}{180^\circ}$, a ratio equal to 1, which is exactly why multiplying by it changes the units without changing the angle. This is the same reasoning that powers the reverse trip, radians to degrees, where you instead multiply by $\frac{180}{\pi}$.
How Is Degrees to Radians Different From Radians to Degrees?
The two conversions are mirror images, and swapping them is the most expensive mistake in this topic.
Degrees to radians: multiply by $\frac{\pi}{180}$. Degrees are the bigger-numbered unit (there are 360 in a circle), so converting to radians usually makes the number smaller and introduces $\pi$.
Radians to degrees: multiply by $\frac{180}{\pi}$. Radians are few (only $2\pi \approx 6.28$ in a circle), so converting to degrees usually makes the number bigger and removes $\pi$.
A fast self-check: if your degrees-to-radians answer came out larger than the original degree number, you used the wrong factor. Converting $60^\circ$ should give the small, tidy $\frac{\pi}{3}$, never a number in the dozens. Keep the two degree conversions on opposite mental shelves and the direction takes care of itself.
What Are the Common Degree-to-Radian Conversions?
Memorising a handful of these speeds up trigonometry, because these angles recur constantly. Each is just the degree value times $\frac{\pi}{180}$, simplified.
Degrees | Multiply by π/180 | Radians |
|---|---|---|
0° | $0 \times \frac{\pi}{180}$ | $0$ |
30° | $30 \times \frac{\pi}{180}$ | $\frac{\pi}{6}$ |
45° | $45 \times \frac{\pi}{180}$ | $\frac{\pi}{4}$ |
60° | $60 \times \frac{\pi}{180}$ | $\frac{\pi}{3}$ |
90° | $90 \times \frac{\pi}{180}$ | $\frac{\pi}{2}$ |
120° | $120 \times \frac{\pi}{180}$ | $\frac{2\pi}{3}$ |
180° | $180 \times \frac{\pi}{180}$ | $\pi$ |
270° | $270 \times \frac{\pi}{180}$ | $\frac{3\pi}{2}$ |
360° | $360 \times \frac{\pi}{180}$ | $2\pi$ |
The pattern worth noticing: the denominator shrinks as the angle grows, and every multiple of $30^\circ$ or $45^\circ$ lands on a clean fraction of $\pi$.
Examples of Degrees to Radians
The examples move from a clean special angle to a decimal and a real-world sweep, with one deliberate wrong turn.
Example 1
Convert 45° to radians.
Multiply by $\frac{\pi}{180}$:
$$45 \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4}$$
Final answer: $\frac{\pi}{4}$ radians.
Example 2
Convert 90° to radians.
The tempting move, if the two conversions have blurred together, is to multiply by $\frac{180}{\pi}$:
$$90 \times \frac{180}{\pi} = \frac{16200}{\pi} \approx 5157$$
Stop and look. An angle of $90^\circ$ is a right angle, a quarter turn, so in radians it must be a small piece of $2\pi \approx 6.28$. An answer of 5157 radians is over 800 full turns, which is absurd. The factor was upside down.
Use $\frac{\pi}{180}$:
$$90 \times \frac{\pi}{180} = \frac{\pi}{2} \approx 1.571$$
Final answer: $\frac{\pi}{2}$ radians.
Example 3
Convert 120° to radians.
$$120 \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3}$$
Final answer: $\frac{2\pi}{3}$ radians.
Example 4
Convert 30° to radians as a decimal, rounded to three places.
$$30 \times \frac{\pi}{180} = \frac{\pi}{6}$$
$$\frac{\pi}{6} = \frac{3.14159}{6} \approx 0.524$$
Final answer: approximately $0.524$ radians.
Example 5
Convert 210° to radians.
$$210 \times \frac{\pi}{180} = \frac{210\pi}{180}$$
Simplify the fraction by dividing top and bottom by 30:
$$\frac{210\pi}{180} = \frac{7\pi}{6}$$
Final answer: $\frac{7\pi}{6}$ radians.
Example 6
A Ferris wheel car moves through 75° of a turn. Express that rotation in radians.
$$75 \times \frac{\pi}{180} = \frac{75\pi}{180}$$
Divide top and bottom by 15:
$$\frac{75\pi}{180} = \frac{5\pi}{12}$$
Final answer: $\frac{5\pi}{12}$ radians.
Why Do Radians Matter More Than Degrees in Higher Math?
"The calculator was never counting in degrees."
Degrees are a human convention: 360 is a Babylonian choice, convenient because it divides so many ways. Radians are a mathematical necessity. When an arc length equals the radius, the angle is one radian, so radians tie angle directly to length. That link is why calculus formulas stay clean only in radians: $\frac{d}{dx}\sin x = \cos x$ is true when $x$ is in radians and false when it is in degrees, where an ugly factor of $\frac{\pi}{180}$ leaks in. The radian is the SI unit of angle for exactly this reason, as documented by Britannica.
Where the conversion earns its keep:
Physics. Angular velocity, simple harmonic motion, and rotational energy all assume radians; feeding degrees in silently corrupts every result.
Computer graphics and games. Trigonometric library functions expect radians, so any degree-based angle from a designer is converted first.
Engineering and signal processing. Wave equations and phase angles live in radians, where the period of a sine wave is a clean $2\pi$.
What Are the Most Common Mistakes Converting Degrees to Radians?
Mistake 1: Multiplying by 180/π instead of π/180
Where it slips in: when both conversions are freshly learned and the two factors blur.
Don't do this: using $\frac{180}{\pi}$ to go from degrees to radians.
The correct way: degrees to radians multiplies by $\frac{\pi}{180}$; radians to degrees multiplies by $\frac{180}{\pi}$. The first instinct after seeing both formulas is to grab whichever one is on top of memory; the fix is the size check, since a radian answer must be smaller than its degree count.
Mistake 2: Converting a decimal too early and losing exactness
Where it slips in: problems that later plug the angle into an exact trig value.
Don't do this: writing $60^\circ = 1.047$ radians and carrying the rounded decimal forward.
The correct way: keep the answer as $\frac{\pi}{3}$ until a decimal is genuinely required. The rusher, who reaches for the decimal button immediately, loses the clean $\pi$-fraction that makes the next step easy.
Mistake 3: Forgetting to simplify the fraction
Where it slips in: large angles like 210° or 300°.
Don't do this: leaving $\frac{210\pi}{180}$ as the final answer.
The correct way: divide numerator and denominator by their common factor to reach $\frac{7\pi}{6}$. An unsimplified fraction is not wrong, but it hides the recognisable special angle.
Conclusion
To convert degrees to radians, multiply by $\frac{\pi}{180}$, because $180^\circ = \pi$ radians.
The factor $\frac{\pi}{180}$ is a ratio equal to 1, so it changes the unit without changing the angle.
Radians to degrees is the reverse, multiplying by $\frac{180}{\pi}$; confusing the two is the top mistake.
Keep answers as exact multiples of $\pi$ and simplify the fraction to reveal special angles.
Higher math, physics, and graphics all run on radians, which is why the conversion matters.
To strengthen angle work with a teacher, explore Bhanzu's trigonometry tutor or a high school math tutor, or browse math classes online.
Practice the six conversions, then rebuild the common-angle table from memory using only $\times \frac{\pi}{180}$. If a direction feels uncertain, come back to the size check: radians should be the smaller number. To practise conversions with a Bhanzu trainer, book a free demo class.
Read More
Was this article helpful?
Your feedback helps us write better content