A B C

a. 

This is not a net for a cube since it would not close.

b. 

This is not a net for a cube since there are not enough faces.

c. 

Yes. Fold and check.

d. 

Yes. Fold and check.

Problem B2

First we declare two nets identical if they are congruent; that is, they are the same if you can rotate or flip one of them and it looks just like the other. For example:

These can be considered identical since a 180° rotation turns one onto the other.

One way of classifying the nets is according to the number of squares aligned (four in the example above).

If five or six squares are aligned, we cannot fold the net into a cube since at least two squares would overlap and the cube would not be closed. So valid nets are to be found among those nets that have four, three, or two squares aligned. Eliminating redundancies (i.e., taking just one net from each pair of equivalent nets), we can come up with the following 11 valid nets:

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Problem B3

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Problem B4

If we assume that two nets are equivalent if one can be rotated or flipped to overlap with the other one, there are six possible nets for a square pyramid.

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Problem B5

A net for a cylinder might look like this:

It will consist of a rectangle and two congruent circles. One pair of the rectangle's sides must have the same length as the circumference of the two circles. The two circles must attach to those two sides of the rectangle, though they need not be positioned exactly opposite each other as shown here.

You can test this by unfolding a layer from a roll of paper towels. Results differ from the rectangle most people will predict.

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Problem B6

Predictions may vary. Many people are surprised to find that the net will look like a circle and a sector of a larger circle:

As with the cylinder, the circumference of the circle must equal the length of the arc of the given sector.

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Problem B7

The shape we get is a sector of a circle. This is because every point on the bottom edge of the cone-shaped object or party hat is equidistant from the top point. This is also true of a sector of a circle because all the radii of the same circle are the same length.

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Problem B8

A larger sector would increase the area of the base and decrease the height of the cone, while a smaller sector would decrease the area of the base and increase the height. All the radii of the same circle are the same length.

Note that you can test this out with the party hats: You can cut a piece off one of them from the center to its edge, then refold it and compare it to the original. You can also tape together two unfolded party hats of the same size and then refold to see what kind of cone (hat) you get.

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