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I've had this question on my mind for quite a while and looking at the web, I couldn't find an answer to this question.

I've had a lot of physics experiments involving springs, unintentionally dismantled a hair clipper once, helped my dad repair his car, and all of them have very identical springs.

Here's an image of a variety of springs that are used for multiple purposes:

enter image description here

So, what's with the spiral/helical shape? And why are they "curled" in circle-like shapes. Why not triangle, square or any other geometric shape? What exactly is the physics behind this.

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I think you should be more specific in your question, indeed if the question is why the section of the springs is often circular, I think that this has just to do with the fabrication process. To build a spring with a circular shape is faster and cheaper that building a spring with, for example, a triangular section at industrial level. However I'm pretty sure that also a spring with a triangular section would work. While if your question is why the springs have an helical (stretched) shape, my answer is that there are a lot of variety of springs and not all have this shape. Maybe these are the most common, but there are some other. For example this is the spring of a mechanical watch, it is flat: enter image description here This is a leaf spring, used in automotive: enter image description here I mean you can project the shape of a spring according to what you need to do,and the helical shape is good just for some applications, not for all.

Landau
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The answer is that to fit a long length of spring into a small physical space you need somehow to curl the spring up, and a spiral is a good way of doing that. The reason that the spirals are round is twofold: first of all making round things is much easier than making things which are odd shapes, and secondly the stresses in materials are higher for smaller radii of curvature (this is why, for instance, cracks propagate, and why the square windows in the original Comet were so catastrophic). So if you want to make something strong you want to make the smallest radius of curvature as large as possible, and the way to do that is to make it circular.

Note that springs which don't require to fit in tiny spaces often are not spirals: leaf springs for vehicles are a good example of this.