12.1-12.5 Exploring Solids

NCSCOS: 2.04

New York Planetarium

Epcot Center

Defns. for 3-dimensional figuresDefns. for 3-dimensional figures• Polyhedron – a solid bounded by polygons

that enclose a single region of shape. (no curved parts & no openings!)

• Faces – the polygons (or flat surfaces)

• Edges – segments formed by the intersection of 2 faces

• Vertex – point where three or more edges intersect

Faces, Edges and Vertices

Three dimensional (3D) shapes are defined by the number of faces, edges and vertices (corners) that they have.

VERTEX (plural is vertices)

EDGE

TETRAHEDRON

FACE

ExEx: Is the figure a polyhedron? If so, how many faces, edges, & vertices are there?

Yes,

F =

V =

E =

5

6

9

No, there are curved parts!

Yes,

F =

V =

E =

7

7

12

Types of Solids• Prism – 2 faces (called bases) in

planes. i.e. first example

• Pyramid – has 1 base, all other edges connect at the same vertex. i.e. last example

• Cone – like a pyramid, but base is a circle.

• Cylinder – 2 circle bases.

or

• Sphere – like a ball.

More definitions• Regular polyhedron – all faces are ,

regular polygons. i.e. a cube

• Convex polyhedron – all the polyhedra we’ve seen so far are convex.

• Concave polyhedron –

“caves in”

• Cross section – the intersection of a plane slicing through a solid. Good picture on p.720

5 regular polyhedra• Also called platonic solids.

• Turn to page 721 for good pictures at the top of the page.

• Tetrahedron – 4 equilateral Δ faces

• Cube (hexahedron) – 6 square faces

• Octahedron – 8 equilateral Δ faces

• Dodecahedron – 12 pentagon faces

• Icosahedron – 20 equilateral Δ faces

ExEx: How many faces, edges, & vertices are there?

F =

V =

E =

5

6

9

F =

V =

E =

7

7

12

F + V = E + 2

5 + 6 = 9 + 2

11 = 11

F + V = E + 2

14 = 14