Radians
Radians measure angles by arc length on a unit circle.
About this topic
A radian measures an angle by the arc it cuts on a circle of radius 1: one radian is the angle whose arc length equals the radius, about 57.3°. A full circle is 2π radians, so 180° = π radians. To convert, multiply degrees by π/180, or radians by 180/π.
Radians make formulas such as arc length s = rθ and sector area A = ½r²θ clean, with no extra conversion constants. In calculus the derivative of sin x is cos x only when x is in radians, which is the main reason mathematicians prefer them. Degrees are just a historical choice of 360 divisions.
Example: 60° becomes 60 × π/180 = π/3 radians, and 3π/4 radians becomes 135°. A circle of radius 5 cm with a central angle of 2 radians has an arc length of 10 cm and a sector area of 25 cm². A wheel turning at 3 radians per second moves a point on its rim at 3 times the radius in length per second.
Radians show up in physics for rotation, in engineering for signals and in computer graphics for rotations. Tips: radians are unitless numbers, so do not write a degree sign; memorise π/6, π/4, π/3 and π/2 as 30°, 45°, 60° and 90°; and always check your calculator mode.
Key ideas
- π rad = 180°
- One radian is the angle where arc length equals radius
- Degrees × π/180 gives radians
Quick check
Use cases: where Radians helps
- Homework: review the key idea behind Radians before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Radians without notes.
- Visual learning: play with the interactive lab to see how Radians changes when you adjust its inputs.
- Teaching a friend: explain Radians in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Radians?
Radians measure angles by arc length on a unit circle.
What will I learn in Radians?
π rad = 180°. One radian is the angle where arc length equals radius. Degrees × π/180 gives radians.
Is Radians part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Radians lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Radians?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.