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Private: Learning Math: Geometry
Solids Part B: Nets (30 minutes)
Session 9, Part B
In this part
- Nets with Regular Polygons
- Nets with Circles
Nets with Regular Polygons
Imagine unfolding a cube so that all of its faces are laid out as a set of squares attached at their edges. The resulting diagram is called a net for a cube. There are many different nets for a cube, depending on how you unfold it. Note 2
Problem B1
Which of the following are nets for a cube? Explain how you decided. Try to imagine folding each one, or print them out to explore.
Problem B2
How many different nets for a cube can you make? How did you think about the problem, what method did you use to generate different nets, and how did you check whether or not a new one really was different?
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Video SegmentModeling objects in three dimensions as well as finding their two-dimensional representation is an important part of geometric thinking. Additionally, finding all possible nets of an object can be a challenging task in terms of ability to organize and record your testing methods. After you’ve completed Problems B1 and B2, watch this sequence to further explore different reasoning participants used to find all the nets for a cube. What method did you come up with to organize and record different nets? You can find the first part of this segment on the session video approximately 16 minutes and 52 seconds after the Annenberg Media logo. The second part begins at 18 minutes and 29 seconds after the Annenberg Media logo. |
Problem B3
Find all the possible nets for a regular tetrahedron.
Take it Further
Problem B4
A square pyramid has a square base and triangular faces that meet at a top vertex. Find all the possible nets for a square pyramid.
Problems B1-B4 and the Video Segment problem adapted from Connected Geometry, developed by Educational Development Center, Inc. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
Nets with Circles
Problem B5
What would a net for a cylinder look like? Sketch one, and describe how the various edges are related.
Problem B6
A cone has a circular base and a point at the vertex. If you cut open a cone and unrolled it, what do you think you would have as the net? Predict what the net for a cone will look like, and then draw a picture of your prediction.
Problem B7
Take a party hat or some other cone-shaped object (a “right” cone where the vertex is directly above the center of the circle). Make a single slice up the side of the cone, and unroll it. What shape do you have? Explain why that is the correct net for a cone.
Problem B8
If you took a larger sector of the same circle, how would that change the cone? What if you took a smaller sector?
Notes
Note 2
If you still have a cube built from Polydrons, you can unfold it, but leave all the squares connected to each other. If you are working in a group, compare how everyone unfolded their own cubes to see if others unfolded them the same way.
Solutions
Problem B1
| 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:
Problem B3
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.
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.
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.
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.
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.
