# Cross-Sections of 3D Figures & Revolutions of Solids

Instructor: Sierra Clegg

Sierra has a Bachelor of Science in mathematics and a Master of Arts in Teaching. She has taught high school math for three years.

In this lesson, we will study the connections between 2-dimensional and 3-dimenstional shapes by looking at the similarities between their nets, cross sections, and revolutions.

## Before There was Algebra…

While many students study algebra before geometry, geometry is actually more than a thousand years older than algebra. Before variables and equations ever existed, people could understand and study the basics of geometric objects. The simplest of all these objects is a point. Points put together make lines, lines form 2D polygons and we put polygons together to make 3D shapes like prisms and pyramids. The connections between these 2D and 3D objects are nets, cross sections and revolutions, which we'll explore below.

## 3D Solids and their Nets

A net takes a 3D solid and unfolds it into the 2D shapes that make it up. These can be helpful for identifying cross sections - the 2D shapes formed by slicing into a 3D solid. Consider the nets of prisms in the picture below.

As you can see, they are made up mostly of rectangles (or squares) and two others shapes called the bases. Now consider the pyramids.

Comparatively, these are made up of triangles and only one base. Solids with curves give different types of nets. A cylinder is similar to a prism with two circular bases. To imagine the rest of it, think about the label on a canned food. If you unroll it, you end up with a rectangle. Cones have one base like pyramids and an almost triangular piece as seen below.

Lastly, spheres have different nets, but picture a map of the earth where it seems to be in slices and that's the most common.

## Cross Sections

The shapes that make up nets are, for the most part, the same ones in the cross sections of solids. You can slice into solids any way you want but in this lesson we'll look at the two most common ways--slicing alongside, parallel to the base, or straight down, perpendicular to the base. In general, slicing parallel gets the same shapes as the base itself. Picture biting the tip off of an ice cream cone. You create a circle just like the larger opening. For prisms and pyramids the shape depends on what the base is, but you can tell what it will be from the net.

Slicing perpendicular to the base will get you the other shape found in the net. For prisms and cylinders, it will be a rectangle and for pyramids and cones it is a triangle. Spheres don't quite match their nets but anyone who has sliced into an orange knows it's cross section is a circle.

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