how to measure the arc length?

Question

We're told to measure angles in radians, θ = arc length/radius. Therefore, 1 radian occurs when the radius of the circle is equal to the arc length subtending the angle you're looking to measure (θ). I am probably overthinking this but I have always wondered, HOW do you accurately measuring the arc length if it is curved? I thought to accurately measure distance you would have to take the arc length and straighten it out? Does it have to do with parameterization or the calculation of circumference of a circle? If you can help shed any light on how radian measure works that'd be amazing thank you!!

Answer 1

You can imagine an arc length calculation as the sum of infinitely many infinitesimally small straight lines $ds$ added together to form the curve. In doing so, we sort of avoid the need to rigorously define a radian. From the Pythagorean Theorem, we have

$$ds=\sqrt{dx^2+dy^2}=\sqrt{1+\left(\frac{dy}{dx}\right)^2}$$

Therefore, the arc length of the whole curve defined by $f:[a,b]→\Bbb{R}^2$ is $$s=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$$

See here for more.

Answer 2

To measure practically, use a tailor's flexible tape bending it along the arc so there are no gaps or intersections.

To calculate.. measure the angle in degrees subtended at center of circle, convert to radians and then multiply with radius of circle.