12.1 Exploring Solids 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

Ex Ex: Is the figure a polyhedron? If so, how many faces, edges, & vertices are there? Yes, F = V = E = No, there are curved parts! Yes, F = V = E = 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 796 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

Thm: 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 Ex: How many faces, edges, & vertices are there? F = V = E = F = V = E = 7 12 F + V = E = = 11 F + V = E = = 14

12.2 – Surface Area of Prisms And Cylinders

Polyhedron with two parallel, congruent bases Named after its base Prism:

Surface area: Sum of the area of each face of the solid

Surface area: Sum of the area of each face of the solid Back Left Top Bottom Front Right

Lateral area: Area of each lateral face

Right Prism: Each lateral edge is perpendicular to both bases

Oblique Prism: Each lateral edge is NOT perpendicular to both bases

Cylinder: Prism with circular bases

Net: Two-dimensional representation of a solid

Surface Area of a Right Prism: SA = 2B + PH B = area of one base P = Perimeter of one base H = Height of the prism H

2. Find the surface area of the right solid. SA = 2B + PH SA = 2(30) + (22)(7) B = bh B = (5)(6) B = 30 P = P = 22 SA = SA = 214m2m2

Surface Area of a Right Cylinder: H SA = 2B + PH

2. Find the surface area of the right solid. cm 2

1. Name the solid that can be formed by the net. Cylinder

1. Name the solid that can be formed by the net. Triangular prism

1. Name the solid that can be formed by the net. rectangular prism