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A square of side length 1 is rotated 360 degrees about one of its vertices. What is the area of the region the square covers while rotating?

I don't know how to visualize this as a geometric shape that I can find the area of. Can someone help?

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  • $\begingroup$ 1) What angle does the diagonal make to the axis of revolution? 2) Are the diagonal and axis in the same plane? $\endgroup$
    – Narasimham
    Commented May 5, 2019 at 15:17

2 Answers 2

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The opposite vertex is the farthest point from the vertex as the centre of rotation. The region is the circle with the diagonal joining the two vertices as a radius.

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1) If the square sides and axis are in the same plane then we have

two cones and two cone frustums connecting them

2) If two far off sides (not connected to sides in contact with axis) are skewed (not in the same plane) then we have

two cones and two segments of one-sheet hyperboloids connecting them.

3) If the plane of square is perpendicular to the rotation axis then we have

a circular sheet radius $\sqrt2$

Images can be uploaded if you so wish.

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