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 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 pyramid Not 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

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 Box How 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