10-1 Vocabulary Face Edge Vertex Prism Cylinder Pyramid Cone Cube Net Cross section

10.1 Solid Geometry Geometry

Defns. 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

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.

Using Properties of Polyhedra CONCEPT SUMMARY TYPES OF SOLIDS Of the five solids below, the prism and the pyramid are polyhedra. The cone, cylinder, and sphere are not polyhedra. prism pyramid cone cylinder sphere

Ex 1: Classify each figure. Name the vertices, edges, & bases are there? _______ v = e = b = __________ ______ v = e = b =

net

Describe the 3 dimensional figure from a net (pg. 655) Check it out Ex. 2a.) Ex. 2b.)

Ex. 3 Describe each cross section on pg. 656 Check it out

Assignment

Cross section

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

Remember the first example? Thm 12.1: Euler’s Theorem The # of faces (F), vertices (V), & edges (E) are related by the equation: F + V = E + 2 Remember the first example? Let’s flashback…

Ex 1: How many faces, edges, & vertices are there? 5 6 9 F + V = E + 2 5 + 6 = 9 + 2 11 = 11 F = V = E = 7 12 F + V = E + 2 7 + 7 = 12 + 2 14 = 14

Ex 2: A solid has 10 faces: 4 Δs, 1 square, 4 hexagons, & 1 octagon Ex 2: A solid has 10 faces: 4 Δs, 1 square, 4 hexagons, & 1 octagon. How many edges & vertices does the solid have? 4 Δs = 4(3) = 12 edges 1 square = 4 edges 4 hexagons = 4(6) = 24 edges 1 octagon = 8 edges F + V = E + 2 10 + V = 24 + 2 10 + V = 26 V = 16 vertices 48 edges total But each edge is shared by 2 faces, so they have each been counted twice! This means there are actually 24 edges on the solid. ( by 2)

Ex 3: A geodesic dome (like the silver ball at Epcot Center) is composed of 180 Δ faces. How many edges & vertices are on the dome? 180 Δs = 180(3) = 540 edges 540  2 = 270 edges F + V = E + 2 180 + V = 270 + 2 180 + V = 272 V = 92 vertices

Assignment