How do I convert an angle from degrees to radians without mixing up the scale?
Quick answer
Multiply the degree measure by π/180, then simplify the fraction: 135° × π/180 = 3π/4 radians. The key scale relationship is 180° = π radians, so degrees and radians describe the same turn using different units. To check your answer, compare the degree angle’s fraction of 360° with the same fraction of 2π radians. [S1]
The question, the details, the next step.
What’s the conversion rule, and how do I avoid accidentally using the factor upside down?
Write the factor so the degree units cancel: multiply the number of degrees by π/180 radians per degree. For example, 60° × π/180 = 60π/180 radians = π/3 radians. The scale behind the rule is 180° = π radians, or 360° = 2π radians for one full turn. [S1]
A useful setup check is to look at the units before calculating: degrees are in the original angle and in the denominator of the factor, so they cancel. If you instead multiply by 180/π, you are setting up the reverse conversion, from radians to degrees.
Can you show the simplification more slowly for an angle that doesn’t immediately reduce to a familiar result?
Try 135°. First attach the conversion factor, then reduce the numbers:
135° × π/180 = 135π/180 radians.
Divide both 135 and 180 by 45: 135/180 = 3/4. So the answer is 3π/4 radians. Keep π in the result; it is part of the exact radian measure, not a label to remove.
For another check, 135° is 135/360 = 3/8 of a full turn. The same fraction of a full radian turn is (3/8) × 2π = 3π/4 radians, which agrees. [S1]
How can I tell whether my answer is a reasonable size, especially when I get a fraction with π?
Compare the angle to a familiar fraction of a full turn. For 30°, the angle is 30/360 = 1/12 of a turn, so its radian measure should be 1/12 of 2π, or π/6. For 300°, it is 300/360 = 5/6 of a turn, giving (5/6) × 2π = 5π/3. These checks catch scale mix-ups: 300° is most of a turn, so an answer smaller than π would not fit that comparison. [S1]
If you need a rough numerical check, 1 radian is about 57.3°. Thus, for example, 30° is a little over half a radian, consistent with π/6 being about 0.52. This is only a reasonableness check; keep the exact answer in terms of π when possible. [S1]
What repeatable steps should I use on homework, and what if the prompt asks for a decimal?
Use this short routine:
- Copy the degree measure, including its sign if there is one.
- Multiply by π/180 so the degree units cancel.
- Reduce the numerical fraction multiplying π.
- Compare the angle’s fraction of 360° with the same fraction of 2π.
- If a decimal is requested, evaluate the simplified expression only after the exact form is clear.
For a copyable worked example with 225°: 225° × π/180 = 225π/180 = 5π/4 radians. Since 225° is 5/8 of a turn, its radian measure should be (5/8) × 2π = 5π/4, so the result passes the turn-size check. [S1]
The cited relationship supports the conversion and these examples; it does not specify how many decimal places a particular assignment requires. Follow the rounding instruction in your problem.
A practical checklist
Put π/180 next to the degree measure so degree units cancel.
Simplify the number multiplying π; keep π in an exact answer.
Check the result by comparing fractions of a full turn.
Use a decimal only when requested, and follow the stated rounding rule.
A few more useful answers
Why is 180° equal to π radians?
A full turn is 360° and 2π radians, so half a turn is 180° and π radians. That relationship gives the conversion factor π/180 radians per degree. [S1]
Should I simplify before or after multiplying by π/180?
Either way is fine if you keep the same value. For clarity, write the full fraction first, then reduce the numerical part multiplying π.
Do I have to convert π to a decimal?
No. Unless a decimal is requested, a simplified expression such as 5π/4 radians is an exact answer. If you do approximate, follow the precision requested by your assignment.
Sources
References used for this guide.
- [S1] OpenStax Precalculus 2e: Angles and Radian Measure openstax.org · Checked