Cuts Through Shapes
Think about 3D shapes by cutting them into pieces.
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.
Explore various ways of thinking about shapes in 3D.
Fold and unfold 3D shapes to see how they fit together.
Fold up nets to make 2D shapes into 3D. Unfold them to see how the faces relate.
What nets can successfully fold up to make a cube?
Explore the faces of a cube and use nets to see how they relate.
Discover how many of these symmetrical solids can be constructed.
Apply the Pythagorean theorem to 3D distances.
How can the shortest distance on the surface of a 3D shape be found?
Puzzle out these 3D distance problems by unfolding the shapes.
Slice 3D shapes into pieces and see what happens.
Think like an MRI machine as you slice through these shapes.
How do cross sections relate to the shape they come from?
Explore the variety of shapes that can be obtained just by slicing up a cube.
Can the cross sections of a solid reveal its full shape?
How many ways are there to cut a 3D solid in half?
Stretch and test your understanding with these cube fraction puzzles.
Find, understand, and prove Euler's formula about the pieces of polyhedra.
Is there a pattern here?
Examine polyhedra that have the same polygons in the same order at every vertex.
Keep cutting solids and see what happens to the shapes as they transform.
Discover the formula that describes the relationship between faces, edges, and vertices.
See why Euler's formula must always be true.
Explore the connections between dual polyhedra and the ways they relate.