Geometry

1. Geometry 3 Dimension

2. Objectives Classify three-dimensional figures according to their properties. Use nets and cross sections to analyze three-dimensional figures.

3. Vocabulary face edge vertex prism cylinder pyramid cone cube net cross section

4. Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces.

6. A cubeis a prism with six square faces. Other prisms and pyramids are named for the shape of their bases.

7. edges: Example 1A: Classifying Three-Dimensional Figures Classify the figure. Name the vertices, edges, and bases. cube vertices: A, B, C, D, E, F, G, H bases: ABCD, EFGH, ABFE, DCGH, ADHE, BCGF

8. edges: Example 1B: Classifying Three-Dimensional Figures Classify the figure. Name the vertices, edges, and bases. pentagonal pyramid vertices: A, B, C, D, E, F base: ABCDE

9. Example 1c Classify the figure. Name the vertices, edges, and bases. cone vertex: N M edges: none base: •M

10. edges: Example 1d Classify the figure. Name the vertices, edges, and bases. triangular prism vertices: T, U, V, W, X, Y bases: ∆TUV, ∆WXY

11. A netis a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face.

12. Example 2A: Identifying a Three-Dimensional Figure From a Net Describe the three-dimensional figure that can be made from the given net. The net has six congruent square faces. So the net forms a cube.

13. Example 2B: Identifying a Three-Dimensional Figure From a Net Describe the three-dimensional figure that can be made from the given net. The net has one circular face and one semicircular face. These are the base and sloping face of a cone. So the net forms a cone.

14. Example 2c Describe the three-dimensional figure that can be made from the given net. The net has four congruent triangular faces. So the net forms a triangular pyramid.

15. Example 2d Describe the three-dimensional figure that can be made from the given net. The net has two circular faces and one rectangular face. These are the bases and curved surface of a cylinder. So the net forms a cylinder.

16. A cross sectionis the intersection of a three-dimensional figure and a plane.

17. Example 3A: Describing Cross Sections of Three-Dimensional Figures Describe the cross section. The cross section is a point.

18. Example 3B: Describing Cross Sections of Three-Dimensional Figures Describe the cross section. The cross section is a pentagon.

19. Example 3c Describe the cross section. The cross section is a hexagon.

20. Example 3d Describe the cross section. The cross section is a triangle.

21. Example 4A: Food Application A piece of cheese is a prism with equilateral triangular bases. How can you slice the cheese to make each shape? an equilateral triangle Cut parallel to the bases.

22. Example 4B: Food Application A piece of cheese is a prism with equilateral triangular bases. How can you slice the cheese to make each shape? a rectangle Cut perpendicular to the bases.

23. Example 4c How can a chef cut a cube-shaped watermelon to make slices with triangular faces? Cut through the midpoints of 3 edges that meet at 1 vertex.

24. triangular prism; vertices: A, B, C, D, E, F; edges: bases: ∆ABC and ∆DEF Question 1 1. Classify the figure. Name the vertices, edges, and bases.

25. Question 2 2. Describe the three-dimensional figure that can be made from this net. square pyramid

26. Question 3 3. Describe the cross section. a rectangle

27. Objectives Draw representations of three-dimensional figures. Recognize a three dimensional figure from a given representation.

28. Vocabulary orthographic drawing isometric drawing perspective drawing vanishing point horizon

29. There are many ways to represent a three dimensional object. An orthographic drawingshows six different views of an object: top, bottom, front, back, left side, and right side.

30. Example 1: Drawing Orthographic Views of an Object Draw all six orthographic views of the given object. Assume there are no hidden cubes.

31. Bottom Example 1 Continued Draw all six orthographic views of the given object. Assume there are no hidden cubes.

32. Example 1 Continued Draw all six orthographic views of the given object. Assume there are no hidden cubes.

33. Example 1 Continued Draw all six orthographic views of the given object. Assume there are no hidden cubes.

34. Example 1B Draw all six orthographic views of the given object. Assume there are no hidden cubes.

35. Example 1B Continued

36. Isometric drawingis a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern.

37. Example 2: Drawing an Isometric View of an Object Draw an isometric view of the given object. Assume there are no hidden cubes.

38. Example 2 B Draw an isometric view of the given object. Assume there are no hidden cubes.

39. In a perspective drawing, nonvertical parallel lines are drawn so that they meet at a point called a vanishing point. Vanishing points are located on a horizontal line called the horizon. A one-point perspective drawing contains one vanishing point. A two-point perspective drawing contains two vanishing points.

41. Helpful Hint In a one-point perspective drawing of a cube, you are looking at a face. In a two-point perspective drawing, you are looking at a corner.

42. Draw a horizontal line to represent the horizon. Mark a vanishing point on the horizon. Then draw a shape below the horizon. This is the front of the . Example 3A: Drawing an Object in Perspective Draw the block letter in one-point perspective.

43. From each corner of the , lightly draw dashed segments to the vanishing point. Example 3A Continued Draw the block letter in one-point perspective.

44. Lightly draw a smaller with vertices on the greyedsegments. This is the back of the . Example 3A Continued Draw the block letter in one-point perspective.

45. Draw the edges of the , using dashed segments for hidden edges. Erase any segments that are not part of the . Example 3A Continued Draw the block letter in one-point perspective.

46. Draw a horizontal line to represent the horizon. Mark two vanishing points on the horizon. Then draw a vertical segment below the horizon and between the vanishing points. This is the front edge of the . Lightly mark a point of the way down the segment, for the lower part of the shape. Example 3B: Drawing an Object in Perspective Draw the block letter in two-point perspective.

47. Example 3B Continued From the marked point and the endpoints of the segment, lightly draw dashed segments to each vanishing point. Draw vertical segments connecting the dashed lines. These are other vertical edges of the .

48. Example 3B Continued Lightly draw dashed segments from the endpoints of each new vertical segment to the vanishing points.

49. Example 3B Continued Draw the edges of the , using dashed segments for hidden edges. Erase any segments that are not part of the .

50. Example 3C Draw the block letter L in one-point perspective. Draw a horizontal line to represent the horizon. Mark a vanishing point on the horizon. Then draw a Lshape below the horizon. This is the front of the L.