A full turn can be measured as 360 degrees or 2π radians. The direction is unchanged; the number and its unit change.
Drag in the picture or use the labelled inputs and sliders. Grid squares span 25 world units. This is a mathematical model running in your browser.
Measure the turn using its circle
A radian is the angle made when an arc’s length equals the circle’s radius. The circumference of a circle is 2πr. Divide by r and one full turn is 2π radians, approximately 6.283185.
That gives the conversion: radians = degrees × π / 180. Going back, degrees = radians × 180 / π. A quarter turn is 90° = π/2 radians; half a turn is 180° = π radians.
Use the π/2 button. The point lies above the origin. Increase the radius: the arc becomes longer, but the angle is still π/2. Arc length divided by radius stays the same. This is why radians are useful when working with circular motion.
A circle has radius 100. You travel an arc of length 100. Have you turned 1 degree or 1 radian?
Show the explanation
1 radian: angle = arc / radius = 100 / 100. That is about 57.296°, much larger than 1°.
Know the function’s convention
JavaScript’s Math.sin and Math.cos take radians. To calculate the sine of 30°, pass 30 × π / 180, giving about 0.5. Passing the bare number 30 asks for the sine of 30 radians, about −0.988. The function cannot guess which unit you intended.
This experiment labels its input in radians and reports degrees alongside it. Enter 1 and compare its direction with π/2. A rounded decimal approximation to π/2 can leave a small horizontal component; the preset uses JavaScript’s π value.