Radian vs Degree: What Is the Difference?

One of them is defined by geometry, the other by agreement

A degree is one three hundred and sixtieth of a full turn. There is no reason for the number beyond history. Nothing in mathematics or physics prefers it, and nothing breaks if you pick a different divisor.

A radian is not like that. The radian is the angle for which the arc length equals the radius, so it is defined by the circle rather than imposed on it:

That second form is printed in the geometry table of all four AP equation sheets, under "Circle", beside $A = \pi r^2$ and $C = 2\pi r$, with a diagram of an arc labelled $s$ and a radius labelled $r$. It is on the sheet, and it holds only in radians.

Two consequences follow, and the whole page is those two consequences worked out.

First: the radian is dimensionless. It is a length divided by a length. That is not a technicality, it is the reason $s = r\theta$ works dimensionally at all: metres on the left, metres times nothing on the right. Push it through and you get $\text{m} \times \text{rad/s} = \text{m/s}$, which is why $v = r\omega$ balances, and $\text{m} \times \text{rad/s}^2 = \text{m/s}^2$, which is why $a_T = r\alpha$ balances. A degree is also dimensionless, and this is where it goes wrong: it is dimensionless and it is not equal to one. One radian equals one. One degree equals $\frac{\pi}{180}$, which is $0.0174533$. So substituting degrees into $s = r\theta$ scales the answer by $57.2958$ while the units still appear to work.

Second: no unit check will ever catch a radian error. Both angular units are dimensionless, so both cancel out of a dimensional analysis identically. The equation looks right, the units come out right, and the number is wrong by a factor of nearly sixty. That is the mechanism behind most silent errors in rotational physics, and it is why the habit has to be conscious.

The CED builds the requirement into its definition rather than leaving it as a note. EK 5.1.A.1: angular displacement is the measurement of the angle, in radians, through which a point on a rigid system rotates about a specified axis. The unit is in the definition of the quantity.

Radian vs degree, side by side

The row about the unit symbols is worth a sentence, because it is easy to misread as an oversight. The four AP sheets each print a box of unit symbols: eight entries on AP Physics 1, eighteen on AP Physics 2, seven on AP Physics C: Mechanics and fifteen on AP Physics C: Electricity and Magnetism. The radian is in none of them, and neither is the degree. That is consistent rather than careless: the boxes list units of dimensional quantities, and an angle is not one.

The row that produces the most wrong answers is the second to last: both units are correct inside a trigonometric function, provided the calculator agrees with you. That is the only place on this page where degrees are as good as radians, and it is also where the most common silent error lives.

The case that separates them: the same theta doing two different jobs

Look at what $\theta$ is asked to do on a single AP equation sheet. The AP Physics 1 symbol key lists $\theta$ twice, as "angle" in the translational column and as "angle or angular position" in the rotational column. Those are two different jobs and they take different units.

The dividing line is not "rotational versus translational". It is whether the angle sits inside a trigonometric function or multiplies a length.

Inside $\sin$ or $\cos$, the function itself absorbs the unit: $\sin 30^\circ$ and $\sin\frac{\pi}{6}$ are both $0.5$, and the only requirement is that your calculator knows which convention you meant. That is why the AP sheets can print a table of trigonometric values in degrees, for $0^\circ$, $30^\circ$, $37^\circ$, $45^\circ$, $53^\circ$, $60^\circ$ and $90^\circ$, without contradicting the rotational block three lines away. All four sheets carry that table and all four print it in degrees.

Multiplying a length, there is no function to absorb anything. The angle's numerical value goes straight into the answer, and only the radian's value makes $s = r\theta$ an identity.

The third case, the constant-angular-acceleration equations, is the one people over-correct. Those contain no length at all, so every term carries the same power of the angular unit and a change of unit multiplies both sides by the same factor. Degrees work in all three of them. You can run an entire angular kinematics problem in degrees per second and degrees per second squared and every answer converts back correctly, right up until you multiply by a radius. Linear vs angular velocity works a turntable problem both ways to show it.

The practical rule that follows: convert to radians before the first equation containing an $r$, and simply converting at the start is always safe.

Why omega and f differ by 2 pi and not by a conversion

Here is the dimensionlessness doing something visible. Angular frequency and ordinary frequency have the same dimensions and are not the same number.

• $f$ is in hertz, which is $\text{s}^{-1}$: cycles per second.
• $\omega$ is in radians per second, which is also $\text{s}^{-1}$, because the radian contributes nothing.

So $\omega$ and $f$ are quantities with identical dimensions that differ by a factor of $6.2832$. Nothing in the units warns you, and no dimensional check will separate them. The $2\pi$ is not a unit conversion; it is the number of radians in the cycle that $f$ counts as one.

A turntable at $33\frac{1}{3}$ revolutions per minute makes the point in four numbers, all describing the same motion:

Two independent conversions are in play and mixing them up is a standard error: dividing by $60$ turns per-minute into per-second, and multiplying by $2\pi$ turns revolutions into radians. Doing only one leaves the answer wrong by $60$ or by $6.2832$.

The period check catches it. All four sheets print $T = \frac{1}{f}$, and the two Physics C sheets also print $T = \frac{2\pi}{\omega} = \frac{1}{f}$. Both routes here give $1.8$ s, so the two numbers describe the same rotation.

The same collapse of units happens elsewhere and is worth recognising. $\text{rad/s}^2$ and $\text{s}^{-2}$ are the same dimensions. $\text{kg}\cdot\text{m}^2/\text{s}$ serves for angular momentum whether you got there through $I\omega$ or through $rmv\sin\theta$. In every one of these the radian is written down for clarity and dropped whenever the algebra needs it to disappear, which is exactly what a dimensionless unit is entitled to do, and exactly what makes it easy to lose track of.

The small-angle approximation only exists in radians

That statement is meaningless in degrees. $\sin 10^\circ$ is $0.1736$ and $10$ is $10$; they are not close. In radians, $10^\circ$ is $0.1745$, and $0.1745$ against $0.1736$ is within half a percent. The approximation is not a fact about small angles. It is a fact about the radian, and it comes from the same definition as everything else on this page: for a small angle the arc and the chord are nearly the same length, and the radian is the arc.

How good it is, in numbers:

The approximation always overestimates, because the arc is longer than the chord.

The AP courses use it in two places and name it in one of them.

AP Physics C: Mechanics prints it as a derived equation. Topic 7.5, Simple and Physical Pendulums, EK 7.5.A.2.ii: for small amplitudes of motion, the small-angle approximation can be applied to the restoring torque, with the derived equations $\sin\theta \approx \theta$ and $\tau = -mgd\theta = I\alpha$. EK 7.5.A.2.iii then says the small-angle approximation and Newton's second law in rotational form yield a second-order differential equation that describes simple harmonic motion, $\frac{d^2\theta}{dt^2} = -\omega^2\theta$.

AP Physics 1 uses the restriction without naming the approximation. EK 7.2.A.1.ii gives the period of a simple pendulum displaced by a small angle as $T_p = 2\pi\sqrt{\frac{\ell}{g}}$, and leaves the reason implicit. So the same physical move appears in both courses with only one of them showing its working.

AP Physics 2 puts it in the exam conventions. The conventions box on the AP Physics 2 Table of Information includes the bullet: the small angle approximation is valid for single- and double-slit diffraction. That is what licenses the small-angle forms on its sheet, $a\left(\frac{y_{\text{min}}}{L}\right) \approx m\lambda$ and $d\left(\frac{y_{\text{max}}}{L}\right) \approx m\lambda$, which replace $\sin\theta$ with $\tan\theta$ and then with $\frac{y}{L}$. Both replacements are the same approximation used twice, and both need radians to be stated as $\sin\theta \approx \tan\theta \approx \theta$.

Which is worth noticing: the small-angle approximation reaches further into the AP courses through optics than through mechanics, and in that setting it is a printed exam convention rather than something you choose to apply.

Calculator mode, and how to catch it in one keystroke

A calculator left in the wrong angle mode is the most self-inflicted error available in this material, because everything about the working is right.

Feed a degree value to a calculator set to radians and it will quietly interpret the number as radians. It returns a real answer, in range, with no complaint. Two examples, both with the same wrong setting:

The torque row is the dangerous one. $15.6$ against $13.5$ is a sixteen percent error, the sign is right, the magnitude is plausible, and nothing in the answer looks wrong. The other two rows are loud because $\sin 40$ and $\cos 40$ in radians happen to land somewhere unhelpful. You cannot rely on the error being loud.

The check takes one keystroke: compute $\sin 30$. If it returns $0.5$, you are in degree mode. If it returns $-0.988$, you are in radians. The AP sheets even hand you the reference value: the trigonometric values table prints $\sin 30^\circ = \frac{1}{2}$ on all four sheets, so the answer is on the page in front of you.

A second check, for the other direction: $\sin 3.14159$ should be about zero in radian mode and about $0.0548$ in degree mode.

Which mode you want depends on the equation, not on the topic:

• Degree mode for geometric angles inside $\sin$, $\cos$ and $\tan$: $W = Fd\cos\theta$, $\tau = rF\sin\theta$, $L = rmv\sin\theta$, $n_1\sin\theta_1 = n_2\sin\theta_2$, projectile launch angles, incline angles, vector components. This covers most of AP Physics 1 and 2, so degree mode is the sensible default for the algebra-based exams.
• Radian mode whenever an angle goes into a trigonometric function as part of an oscillation or wave: $x = A\cos(2\pi f t)$, $x = x_{\text{max}}\cos(\omega t + \phi)$, and anything with $\omega t$ in it. The argument $\omega t$ is in radians because $\omega$ is, so a calculator in degree mode will return the wrong displacement for a perfectly correct equation.

And separately from the mode entirely: any angle that multiplies a radius must be a radian value regardless of what the calculator is set to, because no trigonometric function is involved and the mode setting never comes into it. That is $s = r\theta$, $v = r\omega$, $a_T = r\alpha$ and $\Delta x_{\text{cm}} = r\Delta\theta$.

Where the confusion costs a mark

• Using degrees in $s = r\theta$, $v = r\omega$, $a_T = r\alpha$ or $\Delta x_{\text{cm}} = r\Delta\theta$. Wrong by $57.2958$, and no unit check will find it.
• Using degrees per second in $K = \frac{1}{2}I\omega^2$. Wrong by $57.2958$ squared, which is about $3283$.
• Assuming degrees are forbidden in the angular kinematics equations. They are not. $\omega = \omega_0 + \alpha t$, $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ and $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ contain no length and work in any consistent angular unit.
• Applying $\sin\theta \approx \theta$ with $\theta$ in degrees. The approximation is a statement about radians. In degrees it is not even approximately true.
• Leaving the calculator in the wrong angle mode. Compute $\sin 30$ before you start; degree mode gives $0.5$.
• Working an oscillation in degree mode. $x = A\cos(2\pi f t)$ and $x = x_{\text{max}}\cos(\omega t + \phi)$ have arguments in radians, since $\omega$ is in radians per second.
• Converting revolutions per minute in one step. Divide by $60$ and multiply by $2\pi$. Doing one and not the other is wrong by a factor of $60$ or of $6.2832$.
• Confusing $\omega$ with $f$ because both are per second. They differ by $2\pi$, and they have identical dimensions, so nothing but attention separates them.
• Reading the sheet's degree-based trigonometry table as permission to use degrees everywhere. That table is for geometric angles inside trigonometric functions, which is the one place either unit works.
• Writing an angular answer with no unit at all because the radian is dimensionless. Write "rad", even though it cancels. The reader cannot otherwise tell whether you meant radians, degrees or revolutions.

When it makes no difference, and why that lulls you

Four situations let you use either unit and never find out which you were in.

Any angle that only ever appears inside a trigonometric function. Almost the whole of AP Physics 1 Unit 2 is like this. Incline angles, projectile launch angles, the angle in $W = Fd\cos\theta$, the angle in a vector decomposition: put the calculator in degree mode and the radian never comes up. This is the single largest reason the distinction is learned late, because it holds for the first three or four units of the course.

Any angular kinematics problem with no radius in it. Degrees work throughout, and every answer converts back correctly.

Any ratio or proportional-reasoning question. "If the angular velocity doubles, what happens to the arc length?" needs no unit at all, since the factor cancels.

Any problem stated in revolutions. Counting turns dodges both units, and the conversion only becomes necessary when you multiply by a radius.

Then the distinction arrives, and it arrives all at once, in Unit 5.

• Every equation in the rotational block that contains an $r$. Four of them on the AP Physics 1 sheet.
• Every energy or momentum expression built on $v = r\omega$, which includes $K = \frac{1}{2}I\omega^2$ and $L = I\omega$ even though no $r$ is visible in either.
• Every oscillation written with $\omega t$, which is Unit 7.
• Every small-angle approximation, which is the pendulum in Unit 7 and the diffraction pattern in AP Physics 2.

That is why the safe habit is a conversion rather than a judgement: turn every angular quantity into radians the moment you write it down, and set the calculator to degrees for the geometric angles. Those two rules never conflict, because the geometric angles never multiply a radius and the angular quantities never sit inside a $\sin$.

What the CED asks, and what the sheets print

The radian enters the AP Physics 1 course at Unit 5, Torque and Rotational Dynamics, weighted at 10 to 15 percent of the multiple-choice section. Topic 5.1, Rotational Kinematics, learning objective 5.1.A, describe the rotation of a system with respect to time using angular displacement, angular velocity, and angular acceleration. EK 5.1.A.1 puts the unit inside the definition: angular displacement is the measurement of the angle, in radians, through which a point on a rigid system rotates about a specified axis, with the relevant equation $\Delta\theta = \theta - \theta_0$. EK 5.1.A.2 and EK 5.1.A.3 define the average angular velocity and average angular acceleration, and EK 5.1.A.4.i gives the three constant-angular-acceleration equations. Suggested skills for Topic 5.1 are 1.B, 2.A, 2.D, 3.A and 3.C.

Topic 5.2, Connecting Linear and Rotational Motion, is where the unit becomes compulsory rather than conventional. EK 5.2.A.1 gives $\Delta s = r\Delta\theta$ and EK 5.2.A.2 gives $s = r\theta$, $v = r\omega$ and $a_T = r\alpha$. Suggested skills for Topic 5.2 are 1.C, 2.A, 2.C and 3.B.

The small-angle approximation is named in AP Physics C: Mechanics, Unit 7, Oscillations, at Topic 7.5, Simple and Physical Pendulums, EK 7.5.A.2.ii, with the derived equations $\sin\theta \approx \theta$ and $\tau = -mgd\theta = I\alpha$, and EK 7.5.A.2.iii giving $\frac{d^2\theta}{dt^2} = -\omega^2\theta$. Suggested skills for Topic 7.5 are 1.B, 2.A, 2.B, 3.A and 3.B. AP Physics 1 restricts the simple pendulum to a small angle at EK 7.2.A.1.ii without naming the approximation. AP Physics 2 lists it as an exam convention on its Table of Information, for single-slit and double-slit diffraction.

On the sheets, read off rendered images of all four appendices rather than a transcription:

• All four print $s = r\theta$, in the geometry table under "Circle", with a figure showing the arc $s$, the radius $r$ and the angle $\theta$, and a symbol key giving $s$ as arc length and $\theta$ as angle. This line is easy to miss because it sits in the geometry table rather than the mechanics block, and a transcription that covers only the physics equations will not have it.
• All four print a table of trigonometric values in degrees, headed "Values of Trigonometric Functions for Common Angles", for $0^\circ$, $30^\circ$, $37^\circ$, $45^\circ$, $53^\circ$, $60^\circ$ and $90^\circ$.
• All four print $v = r\omega$ and $a_T = r\alpha$ in the rotational block, along with $\Delta x_{\text{cm}} = r\Delta\theta$ and the three constant-angular-acceleration equations.
• All four print $T = \frac{1}{f}$, and the two Physics C sheets also print $T = \frac{2\pi}{\omega} = \frac{1}{f}$, which is the $\omega$ against $f$ relationship in disguise.
• The radian appears in no unit symbols box on any of the four. Neither does the degree.

The symbol keys are worth reading for this too. The AP Physics 1 and AP Physics 2 rotational keys give $\theta$ as "angle or angular position" and $\omega$ as "angular speed"; the two Physics C keys give $\theta$ as "angular position", $\omega$ as "angular frequency or angular speed" and add $\phi$ as "phase angle". In every case the key names the quantity and not the unit, which is one more reason the radian requirement has to be carried in your head.

Related material: Topic 5.1 and Topic 5.2 carry the CED framing, rotational kinematics works the three angular equations, simple harmonic motion is where $\omega t$ starts to matter, and the definitions live at radian, arc length, angular frequency and the small-angle approximation. The AP Physics 1 equation sheet shows the geometry table and the rotational block on adjacent pages.