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Challenge 297 : Solid Slices

If you cut a solid sphere using a single plane cut you will always end up with two new solids, and the surfaces created by the cut will always be circles.

But if you cut a solid cube in the same way, although you will still always end up with two new solids, there is much more variety in the shapes of the surfaces created by the cut.

Which of the following polygons could be the shape of the cut surface when you slice a cube into two?

(i) triangle (equilateral, isosceles or scalene)

(ii) square, oblong, kite, trapezium

(iii) pentagon (if so, can it be regular?)

(iv) hexagon (if so, can it be regular?)

Are there any other polygons that can be obtained as the new surface when you cut a cube into two?

Illustrate your answers clearly! (Remember, it's OK to send in photos.)

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