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.

Interactive
quizzes

22

Concepts and
exercises

155+
  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
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      Introduction to Nets

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

    2. Included with
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      Nets of a Cube

      What nets can successfully fold up to make a cube?

    3. Included with
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      Exploring Cubes

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

    4. Included with
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      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
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      Introduction to Cross Sections

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

    2. Included with
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      Building Intuition

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

    3. Included with
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      Cross Sections of Cubes

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

    4. Included with
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      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
      Brilliant Premium

      Vertices, Edges, and Faces

      Is there a pattern here?

    2. Included with
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      Uniform Vertex Configurations

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

    3. Included with
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      Cutting Solids

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

    4. Included with
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      Euler's Formula

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