Session 9, Part C:
Cross Sections (45 minutes)

In This Part: Slicing Solids | Polygon Shadows

A cross section is the face you get when you make one slice through an object. Below is a sample slice through a cube, showing one of the cross sections you can get.

The polygon formed by the slice is the cross section. The cross section cannot contain any piece of the original face; it all comes from "inside" the solid. In this picture, only the gray piece is a cross section.

Use the Interactive Activity below to work on Problems C1 and C2. For a non-interactive version of the activity, and to work on Problems C3 and C4, you may want to use clay solids and dental floss to derive your answers. Alternatively, you may want to use colored water in plastic solids. Note 3

Problem C1

Try to make the following cross sections by slicing a cube:

 a. a square b. an equilateral triangle c. a rectangle that is not a square d. a triangle that is not equilateral e. a pentagon f. a hexagon g. an octagon h. a parallelogram that is not a rectangle i. a circle

Record which of the shapes you were able to create and how you did it. The Interactive Activity provides you with one way to make each of the shapes that you can, in fact, make as a cross section.

 How may faces does a cube have? Each side of your cross section comes from cutting through a face of your cube.   Close Tip How may faces does a cube have? Each side of your cross section comes from cutting through a face of your cube.

 Problem C2 A couple of the shapes on the list in Problem C1 are impossible to make by slicing a cube. Explain what makes them impossible.

 Problem C3 Find a way to slice a tetrahedron to make a square cross section. How did you do it?

 Problem C4 What cross sections can you get from each of these figures?

 Cross Sections adapted from Connected Geometry, developed by Educational Development Center, Inc. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
 Session 9: Index | Notes | Solutions | Video