Interactive Course

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.

Topics covered

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

Interactive quizzes


Concepts and exercises


Course map

Prerequisites and Next Steps

  1. 1


    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. Introduction to Nets

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

    2. Nets of a Cube

      What nets can successfully fold up to make a cube?

    3. Exploring Cubes

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

    4. Platonic Solids

      Discover how many of these symmetrical solids can be constructed.

    5. Lines Through Cubes

      Apply the Pythagorean theorem to 3D distances.

    6. 3D Shortest Distance

      How can the shortest distance on the surface of a 3D shape be found?

    7. Strings and Ants

      Puzzle out these 3D distance problems by unfolding the shapes.

  3. 3

    Cuts and Cross Sections

    Slice 3D shapes into pieces and see what happens.

    1. Introduction to Cross Sections

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

    2. Building Intuition

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

    3. Cross Sections of Cubes

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

    4. Predicting Solids

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

    5. Halves of Solids

      How many ways are there to cut a 3D solid in half?

    6. Other Fractions of Cubes

      Stretch and test your understanding with these cube fraction puzzles.

  4. 4

    Facet Counting

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

    1. Vertices, Edges, and Faces

      Is there a pattern here?

    2. Uniform Vertex Configurations

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

    3. Cutting Solids

      Keep cutting solids and see what happens to the shapes as they transform.

    4. Euler's Formula

      Discover the formula that describes the relationship between faces, edges, and vertices.

    5. Proving Euler's Formula

      See why Euler's formula must always be true.

    6. Duality

      Explore the connections between dual polyhedra and the ways they relate.