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# Chapter 15

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1. Chapter 15 By: Andrew Shatz & Michael Baker

2. Chapter 15 section 1 Key Terms: Skew Lines, Oblique Two lines are skew iff they are not parallel and do not intersect. (lines ST and UV are skew) A line and a plane, or two planes that are neither parallel nor perpendicular are said to be Oblique.

3. If two points lie in a plane, the line that contains them lies in the plane.

4. If two planes intersect, they intersect a line.

5. Definitions • Two planes, or a line and a plane, are parallel iff they do not intersect.

6. Definitions (continued) • A line and a plane are perpendicular iff they intersect and the line is perpendicular to every line in the plane that passes through the point of intersection. Line AB is perpendicular to lines: BE, BD, and BC

7. Definitions (continued) • Two planes are perpendicular iff one plane contains a line that is perpendicular to the other plane.

8. Chapter 15 section 2 Key Terms: Polyhedron, Faces, Edges, Vertices, Opposite Vertices, Diagonal, Dimensions, Length, Width, Height, Cube

9. Definitions • Polyhedron: a solid bounded by parts of intersecting planes • Rectangular Solid: a polyhedron that has six rectangular faces • Cube: a rectangular solid with equal dimensions

10. Vertex A and Vertex G are an example of opposite vertices

11. l: Lengthw: Widthh: Heightd: Diagonal • The length, width, and height are the dimensions of the rectangular solid • The length of the diagonal of the rectangular solid with the dimensions l, w, h is • The length of a diagonal of a cube with edges of length e is e

12. Chapter 15 section 3 Key Terms: Bases, Prism, Lateral Faces, Lateral Edges, Right Prism, Oblique Prism, Net, Lateral Area, Total Area

13. Prism: A solid geometric figure whose two end faces are similar, equal, and parallel figures, and whose sides are parallelograms.

14. Lateral face Lateral edge

15. Base Lateral Face Lateral Face Lateral Face • The figure above is a net of a triangular prism. • The lateral area of a prism is the sum of the areas of its lateral faces. • The total area of a prism is the sum of its lateral area and the areas of its bases Base

16. If the lateral edges of a prism are oblique to the planes of its bases the prism is an oblique prism. Right Prism Oblique Prism

17. Chapter 15 section 4 Key Terms: Altitude, Volume, Cross Section, Cavalieri’s Principle,

18. An altitude of a prism is a line segment that connects the planes of its bases and that is perpendicular to both of them.

19. The volume of an object is the amount of space that it occupies. Volume is measured using cubed units. For example, A cross section of a geometric solid is the intersection of a plane and the solid.

20. Cavalieri’s Principle Consider two geometric solids and a plane. If every plane parallel to this plane that intersects one of the solids also intersects the other so that the resulting cross sections have equal areas, then the two solids have equal volumes.

21. Extra Lesson Finding values of Prisms and Pyramid using limited information.

22. Things to Know About Prisms • Base area x lateral edge = volume • Base area + lateral area = total area • You can use the equations above to find base area, lateral edge, lateral area, volume • To find base edges using base area you must know the shape of the base and apply the correct area formula. • For example, if a square has a base area of 50, then the base edges would equal or 5 because the area formula of a square is

23. Things to Know About Pyramids • 1/3 (Base area x altitude) = volume • Base area + lateral area = total area • Using these four formulas you can find base area, altitude, volume, lateral area, total area, base edge, slant height, and lateral edge of a pyramid. • If there is not a sufficient amount of information to plug into the formulas for prisms and pyramids, drawing a picture and labeling what is given to you usually helps.

24. Chapter 15 section 5 Key Terms: Pyramid, Base, Lateral Faces, Lateral Edges, Apex, Altitude

25. Pyramids A pyramid is a polyhedron in which the base is a polygon, and the sides are triangles leading up to the apex. The pyramid lies on its base. All other faces are lateral faces, and the edges they intersect at are lateral edges. All of the lateral edges meet at a single point called the apex.

26. The Altitude is a perpendicular line segment connecting the apex to the to the same plane as the base. Note that the altitude does not necessarily have to meet the base. Volume: Surface Area:

27. Chapter 15 section 6 Key Terms: Cone, Cylinder, Axis, Right Cone, Oblique Cone, Base, Lateral Surface

28. Cones A cone is a figure that has a circular base, and all segments lead up to its apex. The line segment connecting the apex to the center of the of the base is called the axis. If the axis is perpendicular to its base, then the cone is a right cone. If it is oblique to its base, then it is an oblique cone.

29. Types of Cones Right Cone Oblique Cone Volume: Surface Area:

30. Cylinders A cylinder is a figure that has two congruent and parallel circular bases that are connected. A cylinder has two flat bases, and a curved side which is its lateral surface. The axis connects the centers of the bases. A cylinder be right or oblique depending on its axes with respect to its bases. Volume: Surface Area:

31. Chapter 15 section 7 Key Terms: Sphere, Center, Radius, Diameter

32. Spheres A sphere is a set of all points in space that are at a given distance from a given point. The center, radius, and diameter have the same meanings for spheres as they do for circles. Volume: Surface Area:

33. Chapter 15 section 8 Key Terms: Similar, Surface Area, Square, Volumes, Cube

34. Similar Solids Two geometric solids are similar if they have the same shape. Theorem 85: The ratio of the surface areas of two similar rectangular solids is equal to the square of the ratio of any pair of corresponding dimensions. Theorem 86: The ratio of the volumes of two similar rectangular solids is equal to the cube of the ratio of any pair of corresponding dimensions.

35. Chapter 15 section 9 Key Terms: Regular Polyhedron, Tetrahedron, Octahedron, Icosahedron, Cube, Dodecahedron

36. The Regular Polyhedra A regular polyhedron is a convex solid having faces that are congruent regular polygons, and having an equal number of polygons that meet at each vertex. Examples:Equilateral Triangles Icosahedron, 20 Faces Tetrahedron, 4 Faces Octahedron, 8 Faces

37. More Examples! Squares Cube, 4 Faces Regular Pentagons Dodecahedron, 12 Faces

38. Insight! A huge part of this chapter is remembering the various formulas. Without knowing them, it will be nearly impossible to complete problems. Remembering the regular polyhedra is also a large part. The main thing is knowing how many sides there are, and what shapes the faces are. Lastly, knowing the vocabulary of the chapter will make it immensely easier to solve problems.