## 3D Geometry

Entering the 3rd dimension!

Cuts Through Shapes

Surfaces of Shapes

Pieces of 3D

Introduction to Nets

Nets of a Cube

Exploring Cubes

Platonic Solids

Lines Through Cubes

3D Shortest Distance

Strings and Ants

Introduction to Cross Sections

Building Intuition

Cross Sections of Cubes

Predicting Solids

Halves of Solids

Other Fractions of Cubes

Vertices, Edges, and Faces

Uniform Vertex Configurations

Cutting Solids

Euler's Formula

Proving Euler's Formula

Duality

### Course description

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. To succeed at this course, you should already have some familiarity with the basics of 2D geometry. Additionally, some algebra is used in this course, but nothing beyond the level of Algebra I.

### Topics covered

- Cross Sections
- Dissecting Shapes
- Distance in 3D
- Dual Polyhedra
- Euler's Formula
- Folding
- Nets
- Paths on a Surface
- Platonic Solids
- Polyhedra