What exactly is “width” in geometry?

I had never thought I would ever wonder what “width” is until my 12-year old daughter came home one day and told me that her math teachers (not just one but two) told her that “width” in geometry is the vertical side of a rectangle. That to me was like saying up is down and down is up. How could…

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The vicinity geometry in a strip of width w. The red lines illustrate

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Width