Radians
Angle measured by arc length on a unit circle; 2π rad = 360°.
Key idea
Angle measured by arc length on a unit circle; 2π rad = 360°.
In depth
A radian is the angle at the centre of a circle that cuts off an arc equal in length to the radius. Because it comes from the circle itself, formulas become simple: arc length s = rθ and calculus derivatives of sin and cos are clean. A full turn is 2π radians.
Worked example
An arc of length 6 on a circle of radius 3 subtends 6/3 = 2 radians (about 114.6°).
Formula
Real-world uses
- Measuring angular speed in motors and wheels
- Writing formulas in calculus and physics
Common mistake
Thinking a radian is a small angle like a degree. One radian is about 57.3°.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Radians?
Angle measured by arc length on a unit circle; 2π rad = 360°. It is a topic at the intermediate level in Trigonometry.
What is the formula for Radians?
The key formula is rad = deg × π/180.
Where is Radians used in real life?
Real-world examples: Measuring angular speed in motors and wheels; Writing formulas in calculus and physics.
What should I learn before Radians?
Make sure you are comfortable with Angle Measure: Degrees, the previous topic in Trigonometry.
What is a common mistake with Radians?
Thinking a radian is a small angle like a degree. One radian is about 57.3°.
How can I practise Radians?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Radians?
Move on to Converting Degrees and Radians next.
Related intermediate topics in Trigonometry
- Converting Degrees and Radians – Multiply by π/180 or 180/π.
- Sine Ratio – Opposite over hypotenuse.
- Cosine Ratio – Adjacent over hypotenuse.
- Tangent Ratio – Opposite over adjacent.
- Reciprocal Trig Functions – Cosecant, secant and cotangent.
- Solving Right Triangles – Find all sides and angles using ratios and the Pythagorean theorem.