How Does the Equation for Arc Length Work?

I would suggest thinking about how it works visually/graphically. It should make sense that way.

The formula stands for the arc length being equal to the radius multiplied by the angle (in radians).

The equation can be written as s = r θ

Whole Circle & Half a Circle

If you found the circumference of a circle, it would be 2 π r.

If you found half the circumference of a circle, it would be π r.

The angle included in a whole circle is 2 π, the angle included in half a circle is π.

You multiply the angle (in radians) by the radius to get the arc length for both a circle and half a circle.

And it turns out you multiply the angle (in radians) by the radius to get the arc length for any arc length with an angle.

If the angle is 45 degrees, that’s π/4 radians. So you multiply the angle by the radius to get the arc length. It’s a quarter of a circle.

Regardless of the angle, to get an arc length, you multiply the angle (in radians) by the radius.