1. Objectives: To recognize polyhedra and their parts To visualize cross sections of space figures

2.  Polyhedron  a three-dimensional figure whose surfaces are polygons  Face  name for a side of a polyhedron  Edge  a segment that is formed by the intersection of two faces  Vertex  a point where three or more edges intersect

3. Faces Edge Vertex

4.  Ex: ◦ Identify the vertices/edges/faces of the figure A B C D E FG H

5.  Euler’s Formula ◦ The numbers of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula: F + V = E + 2

6.  Ex: Using Euler’s Formula ◦ Use Euler’s Formula to find the number of edges on a polyhedron with eight triangular faces.

7.  Cross-Section  the intersection of a solid and a plane. You can think of it as a very thin slice of the solid.  Examples of Cross Sections ◦ One slice of bread in a loaf ◦ CAT Scans and MRI’s

8.  Homework #25  Due Tuesday (April 09)  Page 601 ◦ # 1 – 19 all

9.  Objectives: To find the surface area of a prism To find the surface area of a cylinder

10.  Prism  a polyhedron with exactly two congruent, parallel faces called bases. The other faces of the prism are called lateral faces.  Altitude  of a prism is a perpendicular segment that joins the planes of the bases.  Height  of the prism is the length of an altitude.

12.  Lateral Area  of a prism is the sum of the areas of the lateral faces.  Surface Area  the sum of the lateral area of the area of the two bases.

13.  Theorem 11.1 – Lateral and Surface Areas of a Prism ◦ The lateral area of a right prism is the product of the perimeter of the base and the height. L.A. = p · h ◦ The surface area of a right prism is the sum of the lateral area and the areas of the two bases. S.A. = L.A. + 2B

14.  Cylinder  has two congruent parallel bases, just like a prism. However, the bases of a cylinder are circles.  Altitude  of a cylinder is a perpendicular segment that joins the planes of the bases  Height  of a cylinder is the length of an altitude

16.  Lateral Area  visualize “unrolling” the curved surface of the cylinder. Imagine taking the label off of a water bottle. The area of the resulting rectangle is the lateral area.  Surface Area  of a cylinder is the sum of the lateral area and the areas of the two circular bases.

18.  Homework # 26  Due Wednesday (April 10)  Page 611 – 612 ◦ #1 – 19 all

19.  Objectives: To find the surface area of a pyramid To find the surface area of a cone

20.  Pyramid  a polyhedron in which one face (the base) can be any polygon and the other faces (lateral faces) are triangles that meet at a common vertex (vertex of the pyramid).  Altitude  of a pyramid is the perpendicular segment from the vertex to the plane of the base.  Height  length of the altitude

21.  Regular Pyramid  a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles.  Slant height (l)  the length of the altitude of a lateral face of the pyramid.

23.  Lateral Area  of a pyramid is the sum of the areas of the congruent lateral faces.  Surface Area  of a pyramid is the sum of the lateral area and the area of its base

25.  Cone  has a pointed top like a pyramid, but its base is a circle  Altitude  a perpendicular segment from the vertex of the cone to the center of its base  Height  the length of the altitude  Slant Height  the distance from the vertex to a point on the edge of the base

29.  Homework #27  Due Thurs/Fri (Apr 11/12)  Page 620 – 621 ◦ # 1 – 21 all

30.  Objectives: To find the volume of a prism To find the volume of a cylinder

32.  Theorem 11.5 – Cavalieri’s Principle ◦ If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume.

33.  Theorem 11.6 – Volume of a Prism ◦ The volume of a prism is the product of the area of a base and the height of the prism. V = B · h h B

34. r h B

35.  Composite Space Figure  a 3D figure that is the combination of two or more simpler figures. ◦ Think of a rocket. It is composed of a conical top and a cylindrical body. The composite volume would be the volume of the cone added to the volume of the cylinder.

36.  Objectives: To find the volume of a pyramid To find the volume of a cone

39.  Homework #28  Due Monday (April 15)  Page 627 – 628 ◦ # 1 – 19 odd  Homework #29  Due Monday (April 15)  Page 634 – 635 ◦ # 1 – 19 odd Quiz Tuesday

40.  Objectives: To find the surface area and volume of a sphere

41.  Sphere  the set of all points in space equidistant from a given point called the center.  Radius  a segment that has one endpoint at the center and the other endpoint on the sphere  Diameter  a segment passing through the center with endpoints on the sphere

42.  When a plane and a sphere intersect in more than one point, the intersection is a circle. If the center of the circle is also the center of the sphere, the circle is called a great circle.  Circumference  of the great circle is the same as the sphere  Hemispheres  two equal halves of a sphere. These are created by a great circle.

47.  Homework #30  Due Monday (April 22)  Page 640 – 641 ◦ # 1 – 21 all

48.  Objectives: To find relationships between the ratios of the areas and volumes of similar solids

49.  Similar Solids  have the same shape, and all their corresponding dimensions are proportional.  Similarity Ratio  the ratio of corresponding linear dimensions of two similar solids. **Any two cubes are similar and any two spheres are similar**

50.  Ex: Identifying Similar Solids ◦ Are the following figures similar? 3 3 2 6 6 4 6 5 1210

53.  A marble paperweight shaped like a pyramid weighs 0.15 lb. How much does a similarly shaped marble paperweight weigh if each dimension is three times as large?

54.  Homework #31  Due Monday (April 22)  Page 648 – 649 ◦ # 1 – 16 all  Test Thursday/Friday