Candy Baskets : Truncated Pyramids
(Mathematically Assisted Design)
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Design and make from poster board open-topped containers in the shape of inverted truncated pyramids, having a regular polygon (of your choice) for a base. Adding handles to the containers makes interesting and fun candy baskets.
Example : Maltese Cross box
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Here is how to design the Maltese Cross box shown above, step by step. (Click to see the image full-sized.):

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The choice of the shape and size is a question of artistic and utilitarian preferences. But there are two questions that the designer may want to know ahead of time.

1. What is the volume of the container?
Answer: If the area of the top is T, and the area of the base is B, and the height is h, then the volume V is V = (B + T + sqrt(B*T))*h/3
Remark If you want to discuss or derive this formula, do it AFTER the task is finished and not during or before the task.

2. Will the container be stable, or will it topple over easily?
Answer: This depends on where the CENTER OF MASS (center of gravity) is located after the container is filled with jelly beans.

About the center of mass
When a container is tilted, as long as a vertical line going through its center of mass passes through its base, the container will straighten itself up, but if the line misses the base, the container will topple over. So, short wide containers are usually stable, and tall and narrow ones are not.
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In the two extreme cases of inverted pyramids, the center of mass is at 1/2 and 3/4 of their height, and in all others it is somewhere in between.
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So in designed containers it will be somewhere in between. But there is a sound engineering principle:
Always expect the worst. This provides a margin of safety.
Thus, in your design, always expect that the center of mass is at 3/4 of its height.
Supplies: colored poster board, scissors, pencils, simple calculators (eg. TI-108)

1. Choose the shape of the base, its size, and the height of the container.

2. Make a side drawing to scale, taking as the width of the base twice the radius of the inscribed circle. (If the base has an even number of sides (4, 6, ...), it is the width of the base. But for odd numbers (3, 5, ...), it will be a little less). Your container will be as stable as a cylinder that is 1 1/2 times taller with a base of the same width. (You may compute the volume at this point to correct the actual size.)
Example The container to the left is as least as stable as the cylinder to the right.

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3. Draw the plan, and make the container.
Remark If the truncated pyramid is very short and broad (its sides are not steep), the container will never truly topple over. It will land on its side, spill its beans, and straighten itself up.

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4. If you put a handle across the top of the container, you can make a candy basket. The baskets in this picture were made by 1st graders. See more pictures of candy baskets in the album.


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