Why do I need to measure a curved object
There are multiple reasons why anyone would need to measure a curved object.
If you work as a plumber and you need to know how wide a hole you need to drill in the counter to pass the pipes through, you would need the radius of the pipe to drill the proper size hole.
If you worked as the designer who is designing his packaging for a soccer ball with open circular ends so the ball can be visible, you would need to know the radius of the ball.
If you were engineering the design of a new car and you were building a mold for the metal curved bumper you would need to know the angle of the bumper.
Most engineering jobs, packaging and construction jobs would at some time have the need to find the radius to complete the requirements of a build or design tasks.