Why Convert Between Degrees and Radians?
An angle can be described using different units. A familiar unit is the degree: one complete revolution is \(360^\circ\). In radian measure, one complete revolution is \(2\pi\) radians. Both units describe the same amount of rotation, but they use different scales.
Converting between the scales is a matter of using their exact equivalence. The key relationship is \(180^\circ=\pi\) radians. From it, we can convert any angle from degrees to radians or from radians to degrees without estimating its size.
Radians are especially important in calculus, so it is useful to become comfortable moving between the two units. This tutorial focuses on the conversion itself and on common rotations. It does not require calculating arc lengths or using coordinates to describe an angle.
Conversion Factors
The equivalence \(180^\circ=\pi\) radians gives two conversion factors. To convert degrees to radians, multiply the degree measure by \(\frac{\pi\text{ radians}}{180^\circ}\). To convert radians to degrees, multiply the radian measure by \(\frac{180^\circ}{\pi\text{ radians}}\). The units cancel in each calculation, leaving the desired unit.
Here \(d\) is the numerical value of an angle measured in degrees, and \(r\) is its numerical value measured in radians. In many calculations the word “radians” is omitted after the number, but the unit is still understood. The factors are reciprocals: \(\frac{\pi}{180}\times\frac{180}{\pi}=1\). That is why one factor reverses the effect of the other.
Why These Factors Work
A conversion factor is a ratio of two equal measures, so its value is \(1\) when both units are interpreted consistently. For example, \(\frac{180^\circ}{\pi\text{ radians}}=1\), because \(180^\circ\) and \(\pi\) radians name the same angle. Multiplying an angle by this ratio changes how it is written, not the angle itself.
The following derivation makes the degree-to-radian rule explicit. Suppose a rotation has numerical measure \(d\) in degrees and numerical measure \(r\) in radians. A full revolution consists of \(360\) degrees or \(2\pi\) radians. The same fraction of a revolution must have the same measure in either unit, so
Multiply both sides by \(2\pi\). Then divide by \(360\), and simplify \(2/360\) to \(1/180\):
Thus a degree measure \(d\) corresponds to \(\frac{d\pi}{180}\) radians. Solving the same relationship for \(d\) gives \(d=\frac{180r}{\pi}\), the radian-to-degree rule. The derivation relies only on the fact that one full revolution is \(360^\circ=2\pi\) radians; the half-turn equivalence \(180^\circ=\pi\) gives the same rules directly.
Converting Degrees to Radians
For a degree measure, multiply by \(\frac{\pi}{180}\), then simplify the numerical fraction if possible. Keep \(\pi\) in the answer for an exact result. For instance, \(\frac{3\pi}{4}\) radians is exact; replacing it with a decimal approximation is unnecessary unless an approximate value is specifically requested.
Worked Example: Convert \(135^\circ\) to Radians
Multiply the degree measure by the factor that cancels degrees:
The degree units cancel. Simplify the fraction by dividing its numerator and denominator by \(45\):
Therefore, \(135^\circ=\frac{3\pi}{4}\) radians. Check by converting back:
The reverse conversion gives the original measure.
The same procedure works when the initial degree measure is negative. The conversion factor is positive, so the sign remains negative. A negative value indicates a rotation measured in the negative direction under the usual signed-angle convention; conversion changes its unit, not its sign.
Worked Example: Convert \(-120^\circ\) to Radians
Multiply by \(\frac{\pi}{180}\), retaining the negative sign:
Divide \(120\) and \(180\) by \(60\):
Thus \(-120^\circ=-\frac{2\pi}{3}\) radians. Check by converting back:
The result has the same negative sign and returns to the starting measure.
Converting Radians to Degrees
To convert a radian measure, multiply by \(\frac{180}{\pi}\). If the angle contains \(\pi\), it will usually cancel, leaving a degree measure. A clear way to avoid reversing the factor is to write the units: radians in the original measure should cancel with radians in the denominator of the factor.
Worked Example: Convert \(\frac{11\pi}{12}\) Radians to Degrees
Multiply by the factor that cancels radians:
Cancel \(\pi\), then divide \(180\) by \(12\):
Thus \(\frac{11\pi}{12}\) radians is \(165^\circ\). Check by converting back:
because \(165\div15=11\) and \(180\div15=12\). The reverse conversion reproduces the original radian measure.
Not every radian measure has \(\pi\) written in its expression. The same rule still applies; in that case, the result may involve a quotient with \(\pi\). Do not assume that every conversion will simplify to an integer number of degrees.
Common Rotations in Both Units
It is helpful to recognize familiar fractions of a full revolution. To find each radian measure, multiply the degree measure by \(\frac{\pi}{180}\). The values below are exact. A “turn” here means one complete revolution, and the fractions describe portions of that revolution.
| Rotation | Degrees | Radians |
|---|---|---|
| One full turn | \(360^\circ\) | \(2\pi\) |
| Half-turn | \(180^\circ\) | \(\pi\) |
| Quarter-turn | \(90^\circ\) | \(\frac{\pi}{2}\) |
| One-third turn | \(120^\circ\) | \(\frac{2\pi}{3}\) |
| One-sixth turn | \(60^\circ\) | \(\frac{\pi}{3}\) |
| One-eighth turn | \(45^\circ\) | \(\frac{\pi}{4}\) |
For example, a quarter-turn is one-fourth of \(360^\circ\), so it is \(90^\circ\). Its radian measure is
The table can also be checked from the radian side. Since one full turn is \(2\pi\), one-fourth of a turn is \(\frac{1}{4}(2\pi)=\frac{\pi}{2}\). Thinking of a rotation as a fraction of a full turn offers a quick check, but the conversion factors work for any angle, not only the entries in the table.
A Common Conversion Mistake
A frequent error is to use \(\frac{180}{\pi}\) when converting degrees to radians. That factor points in the wrong direction: it cancels radians and produces degrees. For example, the incorrect calculation for \(60^\circ\) might be written as
The units remain degrees, so this expression has not converted the measure to radians. The correction is to use the factor with radians in the numerator and degrees in the denominator:
Before doing the arithmetic, check the units. If the starting measure is in degrees, degrees should cancel. If it is in radians, radians should cancel. This simple check catches a reversed factor before it can lead to further errors.
A Reliable Conversion Routine
Decide whether the given measure is in degrees or radians.
Use \(\frac{\pi\text{ radians}}{180^\circ}\) for degrees, or \(\frac{180^\circ}{\pi\text{ radians}}\) for radians.
Cancel units and common numerical factors; retain \(\pi\) when it does not cancel.
Convert it back using the reciprocal factor, or compare it with a familiar fraction of a full turn.
Exact Values and Approximation
For this topic, exact answers are usually the most useful. For instance, \(\frac{5\pi}{4}\) radians is an exact equivalent of \(225^\circ\). A decimal approximation of \(\frac{5\pi}{4}\) is possible, but it does not express the conversion as exactly. Keep \(\pi\) in the answer unless a decimal value is requested.
If a decimal is requested, first perform the exact conversion and then approximate. State the requested precision and round only the final value. Do not round an intermediate quantity if doing so could change the final result. Also distinguish the number from its unit: an angle in degrees and the same angle in radians have different numerical measures.
Summary
The exact equivalence \(180^\circ=\pi\) radians is the starting point for every degree-radian conversion. Multiplying by a ratio of equal angle measures changes the unit without changing the angle. The factor \(\frac{\pi}{180}\) converts degrees to radians; its reciprocal \(\frac{180}{\pi}\) converts radians to degrees.
For a dependable calculation, write the conversion factor with its units, cancel the starting unit, simplify the exact result, and check by converting back. Familiar turns such as a half-turn, quarter-turn, and one-sixth turn provide useful reference values, but the same method applies to any positive or negative angle.
Check Your Understanding
For each question, keep the answer exact unless the question asks for an approximation.
- Convert \(210^\circ\) to radians and check your answer by converting back.
- Convert \(\frac{7\pi}{6}\) radians to degrees and show how \(\pi\) cancels.
- What conversion factor should be used to convert degrees to radians? Explain how the units show that it is the correct factor.
- Write a full turn, a half-turn, and a quarter-turn in both degrees and radians.
- A student converts \(45^\circ\) by multiplying by \(\frac{180}{\pi}\). Identify the error and give the correct conversion.
