Intuition for arc length, angle and radius formula

[gokuls]

I don't understand the intuition/proof of why arc length = arc angle * radius. This may be partially because I don't fully understand the concept of radians, but anyhow please help.

 

[DonAntonio]

I don't understand the intuition/proof of why arc length = arc angle * radius. This may be partially because I don't fully understand the concept of radians, but anyhow please help.

Well, what's the arc length of a complete circle? Just $\,2\pi r\,\,,\,r=\,$ the circle's radius, and since $\,2\pi\,\, radians=360^\circ$ , we

just "adjust" the angle of the arc...

DonAntonio

 

[LCKurtz]

I don't understand the intuition/proof of why arc length = arc angle * radius. This may be partially because I don't fully understand the concept of radians, but anyhow please help.

You have the definition of a radian: 1 radian is measure of the central angle of a circle that subtends an arc of length equal to the radius of the circle.

Now say you have an arc of length $s$ subtended by a central angle $\theta$ measured in radians. Then, since the arc length is proportional to the central angle and a central angle of $\theta= 1$ radian subtends an arc of length $r$,$$
\frac s \theta = \frac r 1$$which gives $s = r\theta$.

 

[Mark44]

You have the definition of a radian: 1 radian is measure of the central angle of a circle that subtends an arc of length equal to the radius of the circle.

To help you (the OP, not LCKurtz) understand this a bit better, sketch a unit circle with its center at the origin. Draw a radius out from the center to the point (1, 0). Draw another radius from the center so that the radius makes an angle of 60° with the x-axis. Connect the two points where the two radii intersect the circle. You should have an equilateral triangle, with all three sides of length 1 and all three angles of measure 60°.
The length along the circle between the two intersection points is a bit larger than 1, since the straight line distance is equal to 1.

If you move the 2nd radius slightly clockwise the right amount, you'll get an arc length that is exactly 1. The central angle between the first radius and the repositioned radius will be a few degrees less than 60°, approximately 57.3°. That's the degree measure of 1 radian.

Now say you have an arc of length $s$ subtended by a central angle $\theta$ measured in radians. Then, since the arc length is proportional to the central angle and a central angle of $\theta= 1$ radian subtends an arc of length $r$,$$
\frac s \theta = \frac r 1$$which gives $s = r\theta$.