In this section students revisit the volume of a cylinder from previous grade levels, and solids of rotation from previous lessons. Then students are introduced to Cavalieri’s Principle: Suppose two solids have equal heights. If, at all distances from the base, the cross-sections of the two solids have equal areas, then the solids have equal volumes. This leads to the idea that the volume of a prism or cylinder with a base area of square units and a height of  units has a volume of  cubic units, regardless of the shape of the base and regardless of whether the solid is oblique.

A pyramid with a square base. The sides of the base are labeled 5. The height, inside the pyramid, is labeled 7. Another pyramid, slanted to the right, with square base. The sides of the base are labeled 5. The height, outside the pyramid, is labeled 7.

If necessary here and in the remainder of the unit, provide students with formulas for the areas of triangles, parallelograms, and circles. 

Note that “area of a circle” is used as shorthand for “the area of the region enclosed by a circle.”

In this unit, students may assume that cylinders or prisms that appear to be oblique are indeed oblique, and those that appear to be right are right. For right cylinders and prisms, right angles will not be marked to indicate that the bases are at right angles to the lateral surfaces.

In this section, students practice spatial visualization. First they do this by examining solids of rotation. Then they investigate cross-sections of a variety of solids. They create physical representations to show that cross-sections of a pyramid may be viewed as dilations of the base for scale factors that are between 0 and 1. Students study the effect of dilation on cross-sections and other two-dimensional figures, establishing that dilating by a scale factor of multiplies areas by . They’re introduced to the equation in this geometric context. They create a graph representing and use it to illustrate the relationship between scaled area and scale factors.

A triangle laying flat with point P above it. Dotted lines connect each vertex of the triangle to point P. Three smaller triangles are drawn with points on the dotted lines, between the first triangle and point P. From bottom to top, the triangles are colored blue, yellow, green, and red.

In this section, students extend their study of scaling to solids. They conclude that dilating a solid by a scale factor of multiplies all lengths by , surface areas by , and volumes by . They work backward from a scaled surface area or volume to find the scale factor involved, which requires the introduction of cube roots. Students create a graph representing and use it to answer questions about how changes in volume affect changes in the corresponding scale factor.

In this section, students combine concepts of dilations, cross-sections, and Cavalieri’s Principle with dissection to derive the formula for the volume of a pyramid or cone. First, they establish that any triangular pyramid whose base has area square units and whose height is units can be combined with two other triangular pyramids of equal volume to form a prism with the same base and height as the original pyramid. Each pyramid has the volume of the prism. Therefore, the volume of the original pyramid is .

Next, students consider a general pyramid, and compare it to a triangular pyramid with equal height and a base of equal area. Because all corresponding cross-sections have equal area, Cavalieri’s Principle applies, and the two pyramids have equal volume. This extends the volume formula to all pyramids and cones, regardless of the particular shape of the base or whether the solid is oblique.

Students then have the opportunity to practice applying volume formulas to a variety of problems. These activities include opportunities for algebraic manipulation and aspects of mathematical modeling.