Geometry: 3-D geometry

MA.912.G.7.1 Describe and make regular, non-regular, and oblique

polyhedra, and sketch the net for a given polyhedron and vice versa.

Block 43

Naming Polyhedra in Mathematics

Polyhedra : naming conventions

• Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), heptahedron (7)

Polyhedra : naming conventions

• Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron.

Polyhedra : naming conventions

• Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron).

Naming common solids in schools

• The names of common solids in schools are a little different

• Space figures are figures whose points do not all lie in the same plane.

• Examples are: polyhedron, cylinder, the cone, and the sphere.

Regular Polyhedra

Regular Polyhedra

• Regular polyhedra are polyhedra in which all the faces are identical regular polygons and all the vertices have the same number of edges.

Regular Polyhedra

There are only 5 types of regular polyhedra:

• the tetrahedron,• the octahedron, • the icohedron,• the cube and• the dodecahedron.

Regular Polyhedra

• Why are there only 5 types of regular polyhedra?

• At a vertex 3, 4 or 5 equilateral triangles, 3 squares, 3 pentagons, etc. can coincide; 3 hexagons would flatten out...

Polyhedrons (Polyhedra)

• Polyhedrons are space figures with flat surfaces, called faces, which are made of polygons.

• Prisms and pyramids are examples of polyhedrons.

Not polyhedra

• Cylinders, cones, and spheres are not polyhedrons, because they have curved, not flat, surfaces.

• A cylinder has two parallel, congruent bases that are circles. A cone has one circular base and a vertex that is not on the base.

• A sphere is a space figure having all its points an equal distance from the center point.

Common three-dimensional solids

Polyhedra:• rectangular prism• pyramidNot polyhedra:• sphere,• cylinder, • cone

Rectangular or square prism

• We can relate some polyhedrons--and other space figures as well--to the two-dimensional figures that we're already familiar with.

• For example, if you move a vertical rectangle horizontally through space, you will create a rectangular or square prism.

Triangular prism

• If you move a vertical triangle horizontally, you generate a triangular prism. When made out of glass, this type of prism splits sunlight into the colors of the rainbow.

Cylinder

• If you move the center of a circle on a straight line perpendicular to the circle, you will generate a cylinder. You know this shape--cylinders are used as pipes, columns, cans, musical instruments, and in many other applications.

Cone

• Cone can be generated by twirling a right triangle around one of its legs. This is another familiar space figure with many applications in the real world.

Sphere

• A sphere is created when you twirl a circle around one of its diameters. This is one of our most common and familiar shapes--in fact, the very planet we live on is an almost perfect sphere! All of the points of a sphere are at the same distance from its center.

Cube

The most famous cube in the world: Rubik’s cube

Cube

Cube

• With six identical squares, 36 hexamines can be formed where each square has at least one edge in common with the other five…

• However, only 11 hexamines correspond to the plane developments of a cube.

• Which of the figures can be folded into a cube

Guided activity

Activity: Building A BoxHow many different nets can you draw that can

be folded into a cube?

Review

• Discuss what are the teacher-specific instructional tools and methods for teaching for this module

• Evaluate the relevant Internet resources designed to reinforce learning