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Generalized length of 3D Cylindrical spiral: NO CALC NEEDED!

In one of my previous problems, I posted finding the length of a spiral stair case. I always wondered how to find the length of a 3D spiral if every segment was congruent to every other. I never thought to visualize it as a cylinder.

Above, you can see the spiral and the cylinder's net. The height of the cylinder is the height of the rectangle and the circumference of the circle is the width. By the pythagorean theorem, the length of one spiral will be \(\sqrt{4\pi^2r^2+h^2}\).

Now, since all parts are congruent, we can multiply it by the degree of turning (1 spiral =360 or \(2\pi\)) relative to one turn. Tus our final formula is \(\sqrt{4\pi^2r^2+h^2}\left(\frac{\theta}{360}\right)\) where \(\theta\) is the degree of turning.

Note by Trevor Arashiro
2 years ago

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And this is why we love math. Thanks for sharing. Arron Kau Staff · 2 years ago

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@Arron Kau Wow, thank you so much. That means a lot to me. :D Trevor Arashiro · 2 years ago

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Nice.... Keep it up dude.. Sanjeet Raria · 2 years ago

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Awesom man . .. Sumit Kumar · 1 year, 9 months ago

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Your solution is great!!! Daniel Osmena · 2 years ago

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Too cool!!!thanks for sharing.totally loved it Tasnia Nowrin · 2 years ago

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Great! Kashif Ahmad · 2 years ago

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