Radian Practice- Conversion Problems and Solutions

What the Hell Is a Radian?

Most students encounter radians in trigonometry or calculus and immediately freeze up. Here's the plain truth: a radian is just a different way to measure angles. Instead of dividing a circle into 360 degrees, radians divide it into 2π units.

One radian is the angle you get when the arc length equals the radius. That's it. Nothing mystical about it.

The conversion formulas you need:

• Radians to Degrees: multiply by 180/π
• Degrees to Radians: multiply by π/180

π ≈ 3.14159, so 180° always equals π radians. Memorize this relationship first.

Common Conversions You Should Know Cold

Stop reaching for your calculator every time you see a multiple of 30° or 45°. These come up constantly:

Learn these. They're the times tables of trigonometry.

Practice Problems

Work through these yourself before peeking at the solutions. That's the only way this stuff sticks.

Problem 1

Convert 5π/3 radians to degrees.

Problem 2

Convert 210° to radians. Simplify your answer.

Problem 3

What is 7π/4 radians in degrees?

Problem 4

Convert 585° to radians. Simplify.

Problem 5

If an angle rotates at 2 radians per second, how many degrees does it rotate in 45 seconds?

Solutions (No Cheating)

Solution 1

5π/3 × (180/π) = 5 × 60 = 300°

The π cancels out. Simple multiplication left.

Solution 2

210° × (π/180) = 210π/180 = 7π/6

Reduce the fraction: divide both 210 and 180 by 30.

Solution 3

7π/4 × (180/π) = 7 × 45 = 315°

180/4 = 45. Then 7 × 45 = 315.

Solution 4

585° × (π/180) = 585π/180

Simplify: divide numerator and denominator by 45.

585 ÷ 45 = 13
180 ÷ 45 = 4

Answer: 13π/4

Solution 5

Step 1: Total radians = 2 × 45 = 90 radians

Step 2: Convert to degrees: 90 × (180/π) = 90 × 180/π = 16200/π ≈ 5156.6°

Or recognize that 90 radians = π/2 × 57.296 ≈ 5156°

Where Students Screw Up

Forgetting to simplify fractions. Your teacher will mark it wrong if you leave 210π/180 instead of 7π/6. Always reduce.

Multiplying instead of dividing. Check which conversion factor you need. Degrees to radians divides by 180. Radians to degrees multiplies by 180.

Confusing the number with the fraction. π radians is 180°, not 3.14°. The decimal approximation comes later. Keep it symbolic until the final step.

Forgetting that radians are unitless. You write "π" not "π radians" in most math contexts. The word "radian" is implied.

Getting Comfortable With Radians

Practice daily. Spend 10 minutes converting random angles back and forth. After a week, you'll do it automatically.

When you're solving trig equations or working with derivatives in calculus, radians are the only sane choice. Degrees will give you wrong answers because the formulas assume radians.

No shortcuts here. Memorize the common values, practice the conversions, and move on. The next topic won't wait.