Notes for Session 8, Part B

 Note 3 The cross section method can be used to find the volume of all prisms (e.g., rectangular, triangular, hexagonal, octagonal) as well as other solids that have congruent parallel bases. In the cross-section method, we find the area of the base (a cross section) of the solid and then multiply that area by the height of the figure. Imagine that you are stacking layer after layer of the base shape on top of itself to build a tower in the shape of the base. This method also works for curved solids such as cylinders that have parallel congruent bases.

 Note 4 Whereas we can use the cross-section method to find the volume of a cylinder, how do we determine the volume of a cone and a sphere? Are the volumes of these shapes in any way related to the volume of a cylinder? They are when the radii of the three solids are identical and when their heights are the same.

 Note 5 This part of the session presents two different methods for determining the actual relationships between the volumes. If you're using manipulatives, it is important to be very careful to make sure the heights and diameters of the cylinder, sphere, and cone are congruent. Sometimes when the clay sphere is flattened into the cylinder, there are holes and gaps, so it appears that the volume of the sphere is greater than it really is. Furthermore, when forming a cone shape that fits into the cylinder, use stiff paper that doesn't have a lot of give to it. Otherwise, you may again have inaccuracies in the relationship between solids. It may be easier to observe the relationships between volume using plastic cylinders, cones, and spheres, as mentioned in the Alternate Experiment. Use water to fill the solids (color it with a drop or two of vegetable food coloring).

 Note 6 This activity asks you to explore the relationship between prisms and pyramids and other cones and cylinders, many of which have the same height and have the same size base. You can use plastic solids and fill them with water, or use paper solids and fill them with rice or sand.