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In this lesson, you will learn how to visualize the 2D cross-sections of 3D shapes

In this lesson, you will learn how to visualize the 2D cross-sections of 3D shapes. Cross Section: the 2 dimensional shape that results from cutting through the solid -.

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In this lesson, you will learn how to visualize the 2D cross-sections of 3D shapes

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  1. In this lesson, you will learn how to visualize the 2D cross-sections of 3D shapes Cross Section: the 2 dimensional shape that results from cutting through the solid -

  2. The Great Pyramids of Giza were originally built with a limestone cap at the top. Over the centuries, these caps have eroded away, and the tops of the pyramids are now parallel to the ground. What 2D shape describes the new top of the pyramid?

  3. Parts of a solid Apex • edge, Face, • and vertex • (vertices)

  4. Effect of slicing plane Where plane intersects faces, edges of 2D figure results

  5. Identify characteristics of the solid Triangles Square

  6. Vertical slice through apex Vertical, through apex = triangle

  7. Horizontal Cross-section Slices parallel to the base will always be similar to the base

  8. A plane can slice through a solid in any direction cross-sections are always horizontal or vertical

  9. Number of intersected faces = number of edges 4 faces/edges 5 faces/edges

  10. The Great Pyramids of Giza were originally built with a limestone cap at the top. Over the centuries, these caps have eroded away, and the tops of the pyramids are now parallel to the ground. What 2D shape describes the new top of the pyramid? Top of pyramid is square

  11. How do you determine the shape that results from slicing a 3D solid? Ice cream factories test how consistently the ingredients are distributed through each carton by cutting cartons in half for a good view. Describe the 2D figures that result from slicing a carton vertically or diagonally through the top & side.

  12. In this lesson, you will learn how to visualize the 2D cross-sections of cylinders by analyzing if a plane intersects with straight or curved surfaces.

  13. Identify characteristics of the solid Lateral surface (face) Edges Bases

  14. Horizontal Cross-section Distance from center is constant. Therefore it’s a circle r r Slices parallel to base are congruent to base

  15. Vertical Cross-section Vertical slice always creates a parallelogram

  16. Circles are only created by horizontal cross-sections A diagonal cross-section creates a circle

  17. Diagonal Cross-section Distance from center is not constant Actually an ellipse a b

  18. Diagonal Cross-section Types of faces intersected determines types of edges on 2D figure

  19. Diagonal Cross-section Intersects 2 arcs & 2 parallel lines

  20. Ice cream factories test how consistently the ingredients are distributed through each carton by cutting cartons in half for a good view. Describe the 2D figures that result from slicing a carton vertically or diagonally through the top & side. Vertical: rectangle Diagonal: half-moon

  21. How do you know what 2D shapes result from slicing through a cone? What would this cone look like if we slice it diagonally?

  22. Identify characteristics of the solid Lateral surface (face) Apex Base Edge

  23. Double-napped Cone 2 cones sharing 1 apex Applications in algebraic geometry & calculus

  24. Vertical Cross-section Vertical slice through apex always creates a triangle

  25. Vertical Cross-section (cont.) Intersects 1 curved & 1 flat face: parabola 2 curved & 2 flat faces: hyperbola

  26. Horizontal Cross-section Intersecting curved lateral surface • Geometrically similar to base: • circle

  27. Circles are only created by horizontal cross-sections A diagonal cross-section creates a circle

  28. Diagonal Cross-section Distance from center is not constant Actually an ellipse b a

  29. Diagonal Cross-section Types of faces intersected determines types of edges on 2D figure

  30. Diagonal Cross-section Intersects 2 arcs & 2 parallel lines Hyperbola along the 2D plane

  31. Interactive

  32. In this lesson, you have learned how to visualize the 2D cross-sections of cones by analyzing if a plane intersects with straight or curved surfaces.

  33. How do you predict the 3D result of rotating a 2D figure? What 3D shape would result from rotating this rectangle?

  34. Rotating Triangle in 3D Axis bisects triangle Rotation creates a cone

  35. Rectangle: Axis Bisecting Edges perpendicular to axis draw flat faces Edges parallel to axis draw curved surfaces Rotation creates: cylinder

  36. Rectangle: Axis Along Edge Edges perpendicular to axis draw flat faces Edges parallel to axis draw curved surfaces Rotation creates: cylinder

  37. Circle: Axis Bisecting Curved edges draw curved surfaces Rotation creates: sphere

  38. In this lesson you have learned how to predict the 3D results of rotating simple figuresby analyzing the effects of rotations.

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