Radians and Degrees

There are two main ways to measure angles: degrees and radians. Degrees are familiar from everyday life — a right angle is 90 degrees, a full turn is 360 degrees. Radians are the standard in higher math, physics, and engineering. Understanding both systems and how to convert between them is essential.

What Is a Radian?

A radian is defined by the relationship between an arc and a radius. If you take a circle and mark off an arc whose length equals the radius of the circle, the angle that arc subtends at the center is one radian.

Since the circumference of a circle is $2\pi r$, and each radian corresponds to an arc length of $r$, a full circle contains:

$\frac{2\pi r}{r} = 2\pi \text{ radians}$

So a full rotation (360°) equals $2\pi$ radians. A half rotation (180°) equals $\pi$ radians. A quarter rotation (90°) equals $\frac{\pi}{2}$ radians.

The key relationship is:

$180° = \pi \text{ radians}$

Conversion Formulas

Degrees to Radians

Multiply by $\frac{\pi}{180}$:

$\text{radians} = \text{degrees} \times \frac{\pi}{180}$

Radians to Degrees

Multiply by $\frac{180}{\pi}$:

$\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Common Conversions Reference Table

This table lists the angles you will encounter most frequently. Memorizing these — especially the ones marked with an asterisk — saves significant time on tests.

Worked Examples

Example 1: Convert 75° to Radians

$75° \times \frac{\pi}{180} = \frac{75\pi}{180} = \frac{5\pi}{12}$

Simplification: Divide numerator and denominator by their greatest common factor (15).

Answer: $75° = \frac{5\pi}{12}$ radians (approximately 1.309 radians).

Example 2: Convert $\frac{2\pi}{5}$ Radians to Degrees

$\frac{2\pi}{5} \times \frac{180}{\pi} = \frac{2 \times 180}{5} = \frac{360}{5} = 72°$

Notice how $\pi$ cancels. This always happens when converting radians expressed in terms of $\pi$.

Answer: $\frac{2\pi}{5}$ radians $= 72°$.

Example 3: Convert 1 Radian to Degrees

$1 \times \frac{180}{\pi} = \frac{180}{\pi} \approx \frac{180}{3.14159} \approx 57.30°$

This is worth remembering: 1 radian is approximately 57.3 degrees.

Answer: 1 radian $\approx 57.3°$.

Why Radians Matter

You might wonder why we need a second unit when degrees work perfectly well. Radians are preferred in advanced math and science for several reasons:

1. Calculus formulas are simpler. The derivative of $\sin(x)$ is $\cos(x)$ only when $x$ is in radians. In degrees, an extra conversion factor appears.
2. Arc length formula. For a circle of radius $r$, the arc length for angle $\theta$ (in radians) is simply $s = r\theta$. No conversion needed.
3. Physics and engineering. Angular velocity, frequency, and wave equations all use radians naturally.
4. Standardized tests. The SAT and ACT include radian-based problems, and many calculators default to radian mode.

When to Use Each

• Degrees: practical measurement, construction, everyday angles, most trade applications
• Radians: calculus, physics, engineering formulas, unit circle work, standardized test questions that specify radians

The Arc Length Connection

One of the most practical uses of radians is the arc length formula:

$s = r\theta$

where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.

Example: An electrician bends conduit in an arc with a radius of 8 inches through an angle of 45 degrees. How long is the curved section?

First convert to radians: $45° = \frac{\pi}{4}$ radians.

$s = r\theta = 8 \times \frac{\pi}{4} = 2\pi \approx 6.28 \text{ inches}$

Answer: The curved section is approximately 6.28 inches long.

Quick Mental Conversions

These shortcuts help with quick estimation:

• Divide degrees by 60 to get an approximate radian value: $60° \approx 1$ rad (actual: 1.047)
• Multiply radians by 60 to get an approximate degree value: $1$ rad $\approx 60°$ (actual: 57.3°)
• Common fractions of $\pi$: If the radian measure has $\pi$ in it, multiply the fraction by 180 to get degrees. For example, $\frac{\pi}{3} = \frac{1}{3} \times 180 = 60°$.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A radian is the angle subtended by an arc equal in length to the radius — one full circle is $2\pi$ radians
• To convert degrees to radians: multiply by $\frac{\pi}{180}$
• To convert radians to degrees: multiply by $\frac{180}{\pi}$
• Key equivalence: $180° = \pi$ radians (memorize this above all else)
• Arc length formula: $s = r\theta$ (with $\theta$ in radians)
• Use degrees for practical measurement; use radians for calculus, physics, and advanced trig

Return to Trigonometry for more topics in this section.