Step by Step
1
Degrees to radians
Multiply the degree measure by π/180 to convert to radians.
2
Radians to degrees
Multiply the radian measure by 180/π to convert to degrees.
3
Key values worth memorizing directly
180° = π radians, 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6 — memorizing these directly is often faster than recalculating them with the conversion formula every time.
4
Why radians matter: arc length and sector area
The arc length formula s = rθ and the sector area formula A = ½r²θ both require θ to be measured in radians — using degrees in these formulas without converting first gives an incorrect result.
Applied Walkthrough
1
Convert 45° to radians. Multiply by π/180: 45 × π/180 = 45π/180 = π/4 radians.
2
Now use this to find the arc length of a circle with radius 8, for a central angle of 45°. Since the arc length formula requires radians, first confirm the angle is in radians: π/4.
3
Apply s = rθ: s = 8 × (π/4) = 2π ≈ 6.28.
4
If you had mistakenly plugged in 45 (degrees) directly into s = rθ without converting, you'd get s = 8 × 45 = 360 — a wildly incorrect answer, showing why the radian conversion step is essential.
Exam Application
Exams test whether you can convert accurately between degrees and radians in both directions, whether you've memorized the key angle equivalents, and whether you remember that radians (not degrees) are required in the arc length and sector area formulas.
⚠ Common Trap
The most common trap is plugging degree values directly into the arc length (s=rθ) or sector area (A=½r²θ) formulas without converting to radians first — these formulas only work correctly when θ is in radians.
✓ Quick Self-Check
1. How do you convert degrees to radians?
Multiply by π/180.
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2. How do you convert radians to degrees?
Multiply by 180/π.
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3. What is 45° in radians?
π/4.
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4. What units must θ be in for the arc length formula s=rθ to work correctly?
Radians.
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5. What is 180° in radians?
π.
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