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Extending to Three Dimensions

Extending to Three Dimensions

In this chapter, students will use ratios to compare the areas and volumes of similar figures. Given the scale factor between two similar figures, students will be able to express the area and volume scale factors and calculate the missing measures of the figures. Students will first examine circles, developing the formula for the circumference of a circle using similarity and the area formula using dissection arguments.

In relation to circles, cylinders and their properties will be examined. By the use of Cavalieri's Principle, students will be able to justify why the volume formula for cylinders works. Students will also develop the formula for the surface area of a cylinder. In addition to cylinders, some other common three-dimensional figures - cones, pyramids, and spheres - and their properties will be discussed throughout the chapter. The volume formulas for cones, pyramids, and spheres will be developed by Cavalieri's Principle and dissection arguments. The surface area formulas will be developed considering the nets of the solids. Furthermore, students are also expected to explore the relationship between the volumes of cylinders, cones, and spheres, given that they have the same linear measures.

After exploring the properties of the common solids, their cross-sections and the solids generated by rotations of two-dimensional objects will be identified and analyzed. Finally, real-life situations will be modeled using geometric and algebraic reasoning. Some of the examples require the derivation of expressions to represent areas of varying cross-sections, the use of Cavalieri's principle, and the conversion factor, while others require applying the concepts of mass, volume, and density.