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# 3D Geometry

## Entering the 3rd dimension!

Explore the fundamental concepts of three-dimensional geometry: What strangely-shaped 3D pieces can result from slicing up 3D polyhedra with planes? What flat polygons can fold up into 3D shapes? If you're running around on the surface of a cube-world, what's the shortest path between two opposite corners? (The answer to this last one might surprise you.)

In this course, you'll stretch problem-solving techniques from flat figures into a third-dimension and explore some mathematical ideas and techniques completely unique to 3D geometry. For example, you'll investigate and learn how to apply Euler's facet counting formula, a formula which describes a surprising algebraic relationship that relates the number of corners, edges, and faces that any polyhedron can have.

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1. 1

### Introduction

Explore various ways of thinking about shapes in 3D.

1. #### Cuts Through Shapes

Think about 3D shapes by cutting them into pieces.

2. #### Surfaces of Shapes

Fold and fold again to transform 2D shapes into 3D.

3. #### Pieces of 3D

Slice, extrude and transform 3D shapes into different configurations.

2. 2

### Nets and Paths

Fold and unfold 3D shapes to see how they fit together.

1. Included with

#### Introduction to Nets

Fold up nets to make 2D shapes into 3D. Unfold them to see how the faces relate.

2. Included with

#### Nets of a Cube

What nets can successfully fold up to make a cube?

3. Included with

#### Exploring Cubes

Explore the faces of a cube and use nets to see how they relate.

4. Included with

#### Platonic Solids

Discover how many of these symmetrical solids can be constructed.

3. 3

### Cuts and Cross Sections

Slice 3D shapes into pieces and see what happens.

1. Included with

#### Introduction to Cross Sections

Think like an MRI machine as you slice through these shapes.

2. Included with

#### Building Intuition

How do cross sections relate to the shape they come from?

3. Included with

#### Cross Sections of Cubes

Explore the variety of shapes that can be obtained just by slicing up a cube.

4. Included with

#### Predicting Solids

Can the cross sections of a solid reveal its full shape?

4. 4

### Facet Counting

Find, understand, and prove Euler's formula about the pieces of polyhedra.

1. Included with

#### Vertices, Edges, and Faces

Is there a pattern here?

2. Included with

#### Uniform Vertex Configurations

Examine polyhedra that have the same polygons in the same order at every vertex.

3. Included with