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Mom's Basement University

Radians and Degree-Radian Conversion

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn what radians are, why they’re used in trigonometry, and how to convert between degrees and radians.

The Concept

Section titled “The Concept”

You already know degrees. A full circle is 360°, a right angle is 90°, and so on. Degrees work fine for everyday use, but there’s another way to measure angles that turns out to be much more natural for higher math.

Radians measure angles based on the radius of a circle. One radian is the angle you get when the arc length equals the radius. Since the circumference of a circle is 2πr, a full circle contains exactly 2π radians.

The key relationship is simple:

180°=π radians180° = \pi \text{ radians}

From that one fact, you can convert anything:

  • Degrees to radians: multiply by π180\frac{\pi}{180}
  • Radians to degrees: multiply by 180π\frac{180}{\pi}

Common values worth memorizing:

  • 30°=π630° = \frac{\pi}{6}
  • 45°=π445° = \frac{\pi}{4}
  • 60°=π360° = \frac{\pi}{3}
  • 90°=π290° = \frac{\pi}{2}
  • 180°=π180° = \pi
  • 270°=3π2270° = \frac{3\pi}{2}
  • 360°=2π360° = 2\pi

Why bother with radians? Because they make many formulas cleaner. In calculus, the derivative of sin(x) is cos(x), but only if x is in radians. In degrees, you’d need an extra conversion factor everywhere. Radians also connect angle measure directly to arc length, which is useful in physics and engineering.

Worked Example

Section titled “Worked Example”

1. Convert 135° to radians

Multiply by π180\frac{\pi}{180}:

135°×π180=135π180=3π4 radians135° \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4} \text{ radians}

2. Convert 5π6\frac{5\pi}{6} radians to degrees

Multiply by 180π\frac{180}{\pi}:

5π6×180π=5×1806=150°\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = 150°

3. Convert 240° to radians

240°×π180=240π180=4π3 radians240° \times \frac{\pi}{180} = \frac{240\pi}{180} = \frac{4\pi}{3} \text{ radians}

Real-World Application

Section titled “Real-World Application”

Radians are used in:

  • Physics and engineering (angular velocity, rotational motion)
  • Computer graphics and game development (rotations and animations)
  • Navigation and GPS systems
  • Calculus (derivatives and integrals of trig functions are much simpler in radians)
  • Astronomy and satellite communications

Example: a wheel rotating at 3000 revolutions per minute is often described in radians per second for engineering calculations. One revolution = 2π radians, so 3000 RPM = 3000 × 2π / 60 ≈ 314.16 radians per second.

Quiz

Section titled “Quiz”
How many radians are in $180°$?
Convert $90°$ to radians.
Convert $\frac{3\pi}{4}$ radians to degrees.
Why are radians preferred in higher mathematics?
Convert $60°$ to radians.