Radians: Measuring with the Radius

Lesson 3 of 11 · 7 min · Free, no account needed

What you will learn

An angle in radians is the arc length divided by the radius: how many radius lengths the arc is.
A full turn is 2π ≈ 6.28 radians, so 180° = π and 1 radian ≈ 57.3°.
Arc length = radius × angle in radians. That is why code and physics use radians.

Why do programmers and physicists avoid degrees?
Cut a string as long as the wheel's radius. How many of them fit around the rim? Bet with the lever.
Wrap radius strings around the rim.
Press Wrap and count.
Measure angles in strings.
Drag the end of the string along the rim. An arc one radius long is 1 radian. Three goals, then make your own.
That unit is the radian.
angle in radians = arc length ÷ radius. An arc one radius long is 1 radian, about 57.3°.
Half a turn holds π ≈ 3.14 radius lengths, so 180° = π radians.
A full turn is 2π radians = 360°. To change degrees to radians, multiply by π/180.
Put each angle on the unrolled rim.
The string is the rim, straightened out and marked in radians.
Radians = degrees × π/180. For example 120° × π/180 = 2π/3 ≈ 2.09 radius lengths.
Game: roll each wheel to its flag.
A wheel that turns 1 radian rolls forward one radius. Set the turn, then Roll.
A robot arm should rise to sin(90°) = 1. It stops at 0.89.
sin_rad works in radians, like sin in Python and most code. Tap a fix for the first line.
Code's sin and cos take radians. 90 radians is not 90°: it is about 14.3 turns, and sin(90) ≈ 0.89. Multiply degrees by π/180 first.
An arc 3 cm long sits on a circle of radius 2 cm. What is the angle in radians?
angle = arc ÷ radius = 3 ÷ 2 = 1.5 rad, about 86°.
A radian is one radius of arc. A turn is 2π.
Your arc from earlier, as a badge. Drag its end to tune it.

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