Degrees to Radians
The first half of the conversion bridge. Learn the single multiplication step that translates any degree measure into the radian system used by higher mathematics.
Introduction
Now that you understand what a radian physically represents, you need a reliable algebraic tool to convert back and forth between the two systems. This lesson focuses on the direction from degrees to radians.
Past Knowledge
You know that radians equals and that radians is a full rotation.
Today's Goal
Convert any angle measured in degrees to its exact radian equivalent by multiplying by .
Future Success
Every formula in calculus requires radian input. If your calculator is set to degrees and you don't convert, every single derivative and integral will be wrong.
Key Concepts
The Conversion Factor
The master equation from Lesson 3.1 states:
Divide both sides by degrees:
This means that every is equal to radians. To convert any degree angle, simply multiply:
Degrees → Radians Formula
The Two-Step Process
- Multiply the degree value by .
- Simplify the resulting fraction. Look for common factors between the degree number and .
Common Conversions Reference
| Degrees | × π/180 | Radians |
|---|---|---|
Worked Examples
Converting a Standard Angle
Question: Convert to radians.
Step 1: Multiply by the conversion factor.
Step 2: Simplify the fraction.
The GCF of and is .
Final Answer: radians
Converting Beyond 180°
Question: Convert to radians.
Step 1: Multiply.
Step 2: Simplify.
The GCF of and is .
Final Answer: radians
Converting a Negative Angle
Question: Convert to radians.
Step 1: Multiply. The negative sign carries through the multiplication.
Step 2: Simplify. GCF of and is .
Final Answer: radians
Common Pitfalls
Multiplying by the Wrong Fraction
When going from degrees to radians, you multiply by (π on top). When going from radians to degrees (next lesson), you multiply by (180 on top). Mixing these up gives you a nonsensical answer.
✅ Tip: “Degrees to radians? π goes up, degrees go away.”
Not Simplifying the Fraction
An answer of is technically correct but not fully reduced. Always simplify to . Leaving fractions unsimplified will cost points on exams.
Real-Life Applications
Game Engine Rotation APIs
Major game engines like Unity and Unreal Engine perform all internal angle calculations in radians. When an artist sets a rotation to in the visual inspector, the engine silently converts it to behind the scenes. If you write custom rotation scripts using the math API, you must manually convert degrees to radians — or your characters will barely inch forward instead of turning a quarter revolution!
Practice Quiz
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