Radians to Degrees
The return trip across the conversion bridge. When a formula hands you a radian answer, this skill translates it back into the degree language your intuition understands.
Introduction
In the previous lesson you learned to convert degrees into radians. Now we reverse the process. Many calculator outputs and formula results come in radians, so being able to translate back to degrees is essential for checking your work against your geometric intuition.
Past Knowledge
You can convert degrees to radians using .
Today's Goal
Convert any radian measure to degrees by multiplying by .
Future Success
Verifying radian answers in degrees builds the mental “sanity check” habit that prevents catastrophic calculation errors.
Key Concepts
The Reverse Conversion Factor
Starting from the master equation , we now divide both sides by :
So to convert any radian value to degrees:
Radians → Degrees Formula
When π Cancels
Most radian values you encounter will contain as a factor. When you multiply by , the symbols divide out, leaving a clean integer or simple fraction. This is by design — the “nice” degree values like , , all map to radian fractions with in the numerator.
When π Does NOT Cancel
If you see a radian value without (like just the number ), the conversion will produce a decimal degree value:
Worked Examples
Converting a Common Radian Value
Question: Convert radians to degrees.
Step 1: Multiply by .
Step 2: Cancel and simplify.
Final Answer:
Negative Radian Conversion
Question: Convert radians to degrees.
Step 1: Multiply by .
Step 2: Simplify.
Final Answer:
Converting a Decimal Radian (No π)
Question: Convert radians to degrees. Round to one decimal place.
Step 1: Multiply by .
Step 2: Compute the decimal.
Since :
Final Answer:
Common Pitfalls
Using π/180 Instead of 180/π
This is the most common mistake. When converting to degrees, belongs in the numerator so the cancels out and degrees emerge.
❌ (this is radians squared — not a valid unit!)
✅
Real-Life Applications
Navigation and GPS
GPS satellites broadcast latitude and longitude in degrees, but the internal path-planning algorithms on your phone use radians since all distance calculations over a curved Earth rely on the arc length formula (which requires radians). The phone constantly converts between the two systems behind the scenes to display human-friendly degree coordinates while computing radian-based distances.
Practice Quiz
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