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Two instruments that could be used to measure curved lines are a flexible ruler and a caliper. A flexible ruler, also known as a contour gauge, can be molded to the shape of the curved line to measure its length accurately. A caliper, specifically a vernier caliper, can be used to measure the diameter or thickness of a curved object by gently gripping and measuring across the curve.
A curvy line is that is used for rounded angles. This makes it easier to measure.
A tape measurer can be used to measure distance in linear or even around curved surfaces. It is an old tool that has a neat history. See the related link for more information.
Length of any curve is imagined as an infinite sum of the lengths of infinitely small straight sections. If you look at a part of circle under a microscope you cannot tell that it is curved. This idea is used extensively in the branch of mathematics call "integral calculus". On a practical level, you can lay out a circle with a length of wire, then straighten it out to measure its length.
A curvy line is that is used for rounded angles. This makes it easier to measure.