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The shapes below are pyramids. A pyramid is named for the shape of its base. The left shape is a triangular pyramid, the center shape is a square pyramid, and the right shape is a pentagonal pyramid. The sides of all pyramids are triangles.

- As the number of sides in the base of a pyramid increases, what happens to the shape of the pyramid?
- As the number of sides in the base of a pyramid increases, what happens to the volume of the pyramid?

Spectacular Sports manufactures high-quality basketballs. The company packages its basketballs in 1 ft^{3} cardboard boxes. The basketballs fit nicely in the boxes, just touching the sides. To keep the ball from being damaged, Spectacular fills the empty space in the box with foam. How much foam goes into each basketball box?

Start with four identical sheets of paper with familiar dimensions (e.g., 8 1/2 by 11 in.). Use two of the sheets to make two different cylinders by taping either the long sides or the short sides of the paper together. Imagine that each cylinder has a top and a bottom. Take the other two sheets of paper and fold them to make two different rectangular prisms. Imagine that these rectangular prisms also have a top and a bottom. **Note 7**

- Which of the four containers has the greatest volume? Explain your reasoning.

- Which container has the greatest surface area? Explain your reasoning.

- Take a cylindrical and a rectangular container of the same height. Which one has a greater volume?

When folded, what are the dimensions of each of the boxes below? What are the volumes?

Historically, units of measure were related to body measurements. Yet as we saw in Session 2, these measures were most often units of length, such as arm span, palm, and cubit. The cubit, used first by ancient Egyptians, is the distance from a person’s elbow to the tip of the middle finger. The Egyptians standardized the cubit and called their standard measure the royal cubit. The measure of volume in ancient Egypt was a cubic cubit.

- Use your arms to estimate the size of a royal cubit.
- Estimate how many royal cubic cubits are in 1 m
^{3}. How many cubic centimeters are in 1 m^{3}?

The cubic fathom is a unit of measure that was used in the 1800s in Europe to measure volumes of firewood. A fathom is the distance of your two outstretched arms from fingertip to fingertip.

Estimate the size of your cubic fathom. About how much firewood would fit into your cubic fathom? Explain your reasoning.

**Zebrowski, Ernest (1999). A History of the Circle (pp. 77-81). Piscataway, N.J.: Rutgers University Press.**

Reproduced with permission from the publisher. © 1999 by Rutgers University Press. All rights reserved.

**Download PDF File:**

A History of the Circle

To learn more about the research on the role of spatial structuring in the understanding of volume, read the following article:

**Battista, Michael T. (March-April 1998). How Many Blocks? Mathematics Teaching in the Middle School, 3 (6), 404-11.**

Reproduced with permission from

Download PDF File:

How Many Blocks?

Continued

**Note 7**

If you are working in a group, once you have determined which solid has the greatest volume and which solid has the greatest surface area, discuss why these configurations of the original sheet of paper produce these results.

**Problem H1**

- As long as the base is a regular polygon, its shape begins to approximate a circle. The triangular sides become increasingly small. In time, the pyramid becomes indistinguishable from a cone.
- The volume formula is still V = (1/3)Bh, but since the base becomes a circle, the volume becomes V = (1/3)πr2h.

**Problem H2**

If the diameter of the ball is 1 ft., then the radius is 0.5 ft. Also, the height of the box is 1 ft. So the volume of the sphere is (4/3) π (0.5)^{3} = 0.52 ft^{3} (rounded to hundredths using the π key on your calculator). The volume of the box is 1 ft^{3}. To obtain the volume of box that is foam, subtract the volume of the sphere from the volume of the box (1 – 0.52 = 0.48 ft^{3}). So 0.48 ft^{3} is filled with foam. You could convert everything to inches to solve this as well.

**Problem H3**

- The shorter, wider cylinder has the greatest volume. In the formula for the volume of a cylinder, V = πr
^{2}h, notice that the radius is squared and the height is not. Using the larger dimension of the paper as a circumference of the base produces a larger radius, and in turn, an exponentially larger volume, and vice versa: Using a smaller dimension of the paper as a circumference of the base produces a smaller radius, and in turn, an exponentially smaller volume. - Again, the shorter, wider cylinder has the greatest surface area. All will have equal lateral surface area (not including the top and bottom), since the same paper is being used. For total surface area, add the area of the bases — so the problem boils down to which has the largest base area. As described in the solution to part (a), the shorter, wider cylinder has the largest base area.
- The cylindrical container has the greater volume, as long as they both have the same lateral surface. If only the heights are known, then there is no comparison to make — one could have a much larger base area than the other.

**Problem H4**

Box A is 1 by 1 by 6. Its volume is 6 cubic units.

Box B is 1 by 3 by 3. Its volume is 9 cubic units.

Box C is 2 by 2 by 4. Its volume is 16 cubic units.

**Problem H5**

- Answers will vary. Your answers should reveal a royal cubit to be about 52.4 cm or 20.62 in.
- Since a cubit is just over 0.5 m, we would expect there to be just under 2 cubits in each dimension in 1 m
^{3}. This means the volume is just under 8 cubic cubits. Based on the value of 52.4 cm in 1 cubit, there are 6.95 cubic cubits in a cubic meter. There are 1,000,000 cm^{3}in a cubic meter.

**Problem H6**

Answers will vary. A fathom is roughly 2 m, so a cubic fathom is about 8 m^{3}. If one piece of firewood measures 30 cm by 10 cm by 10 cm, one piece of firewood is 3,000 cm^{3}. Since there are 1,000,000 cm^{3} in a cubic meter, then about 333 pieces of firewood fit in 1 m^{3}, and about eight times that (roughly 2,600 pieces) fit in a cubic fathom.