Radians and Degrees

Learn to convert between radians and degrees, two ways of measuring angles.

Advanced25 minLesson

Definition

Radians and degrees are two different units for measuring angles.

Degrees: A full circle is $360°$
Radians: A full circle is $2\pi$ radians

The key relationship:

180° = \pi \text{ radians}

This means:

To convert degrees to radians: multiply by $\frac{\pi}{180}$
To convert radians to degrees: multiply by $\frac{180}{\pi}$

Try it now

How many degrees are in a full circle?

Worked Examples

Convert $45°$ to radians.

Write the conversion formula

Radians $=$ Degrees $\times \frac{\pi}{180}$ → Formula ready

Substitute the value

$45 \times \frac{\pi}{180}$ → $\frac{45\pi}{180}$

Simplify the fraction

$\frac{45\pi}{180} = \frac{\pi}{4}$ → Divide both by 45

Answer: $45° = \frac{\pi}{4}$ radians

Common Mistakes

Forgetting to simplify the fraction after conversion

Why it's wrong: The raw calculation often gives an unsimplified result like $\frac{90\pi}{180}$ .

Correct: Always simplify: $\frac{90\pi}{180} = \frac{\pi}{2}$ . Look for common factors.

Using degrees when calculator is in radian mode (or vice versa)

Why it's wrong: Calculators have different modes. $\sin(90)$ in radian mode gives approximately $0.894$ , not $1$ .

Correct: Always check your calculator mode. $\sin(90°) = 1$ but $\sin(90 \text{ rad}) \approx 0.894$ .

Confusing $\pi$ radians with $180\pi$ degrees

Why it's wrong: Mixing up the conversion direction leads to huge errors.

Correct: $\pi$ radians $= 180°$ (not $180\pi°$ ). Remember: $\pi \approx 3.14$ is the radian value.

Interactive Visual

Unit Circle

Degrees

0°

Radians

sin(θ)

cos(θ)

tan(θ)

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Angle Explorer

0°90°180°

Acute

Drag the blue point or use the slider to change the angle.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

16 problems

Problem 1 of 16

Easy

How many degrees are in a full circle?

Why It Matters

Radians are essential in higher mathematics and science:

Calculus: Derivatives of trigonometric functions only work correctly with radians
Physics: Angular velocity and acceleration use radians per second
Engineering: Circular motion calculations require radians
Programming: Most programming languages use radians for trig functions

While degrees are intuitive for everyday use, radians make mathematical formulas simpler and more elegant!

Real World Applications

Rotating Objects in Video Games

Game developers use radians to rotate characters and objects smoothly on screen.

Example:

A character turning $90°$ to face right rotates $\frac{\pi}{2}$ radians in the code.

1Try It Yourself

A spaceship needs to rotate 270 degrees to face the player.

How many radians should the programmer use?

Step 1: Write the mathematical expression

Convert $270°$ to radians:

Circular Motion in Physics

Scientists measure angular velocity in radians per second for rotating objects.

Example:

A wheel spinning at $2\pi$ radians per second completes exactly one full rotation every second.

2Try It Yourself

A Ferris wheel rotates 60 degrees every 10 seconds.

What is its angular velocity in radians per second?

Step 1: Write the mathematical expression

First convert $60°$ to radians, then divide by time:

Key Takeaways

1 $180° = \pi$ radians is the key relationship
2To convert degrees to radians: multiply by $\frac{\pi}{180}$
3To convert radians to degrees: multiply by $\frac{180}{\pi}$
4Common angles: $30° = \frac{\pi}{6}$ , $45° = \frac{\pi}{4}$ , $60° = \frac{\pi}{3}$ , $90° = \frac{\pi}{2}$
5Radians are preferred in calculus, physics, and programming

Frequently Asked Questions

Radians make calculus formulas much simpler. For example, the derivative of

\sin(x)

\cos(x)

only when

x

is in radians. With degrees, you would need an extra conversion factor.

Glossary

Radian: A unit of angle measurement where the angle subtends an arc equal in length to the radius. One full circle is $2\pi$ radians.
Degree: A unit of angle measurement where one full circle equals $360°$ .
Pi ( $\pi$ ): The ratio of a circle's circumference to its diameter, approximately $3.14159$ .
Reference angle: The acute angle formed between the terminal side of an angle and the x-axis.
Unit circle: A circle with radius 1 centered at the origin, used to define trigonometric functions.

Formula Card

Degrees to Radians

\text{rad} = \text{deg} \times \frac{\pi}{180}

Multiply degrees by pi/180

Radians to Degrees

\text{deg} = \text{rad} \times \frac{180}{\pi}

Multiply radians by 180/pi

Key Relationship

180° = \pi \text{ rad}

The fundamental conversion

Full Circle

360° = 2\pi \text{ rad}

A complete rotation