In this lesson, you will learn how to visualize the 2D cross-sections of 3D shapes Cross Section:...
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Transcript of In this lesson, you will learn how to visualize the 2D cross-sections of 3D shapes Cross Section:...
![Page 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/1.jpg)
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 -
![Page 2: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/2.jpg)
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?
![Page 3: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/3.jpg)
Let’s Review
Parts of a solid
Face, edge,
and vertex(vertices)
Apex
![Page 4: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/4.jpg)
Core Lesson
Effect of slicing plane
Where plane intersects faces, edges of 2D figure results
![Page 5: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/5.jpg)
Core Lesson
Identify characteristics of the solid
SquareTriangles
![Page 6: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/6.jpg)
Core Lesson
Vertical, through apex
=triangle
Vertical slice through apex
![Page 7: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/7.jpg)
Core LessonHorizontal Cross-section
Slices parallel to the base will always be similar to the base
![Page 8: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/8.jpg)
A Common Misunderstanding
A plane can slice through a solid in
any directioncross-sections are always horizontal or vertical
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Core Lesson Number of intersected faces
=number of edges
4 faces/edges
5 faces/edges
![Page 10: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/10.jpg)
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
![Page 11: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/11.jpg)
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.
How do you determine the shape that results from
slicing a 3D solid?
![Page 12: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/12.jpg)
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.
![Page 13: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/13.jpg)
Let’s Review
Identify characteristics of the solid
BasesLateral surface(face)
Edges
![Page 14: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/14.jpg)
Core Lesson
Distance from center is constant.Therefore it’s a
circle
Horizontal Cross-section
rr
Slices parallel to base are congruent
to base
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Core Lesson
Vertical Cross-section
Vertical slice always creates a parallelogram
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A Common Misunderstanding
A diagonal cross-section creates a circle
Circles are only created by horizontal
cross-sections
![Page 17: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/17.jpg)
Core Lesson
Distance from center is not
constantActually an ellipse
Diagonal Cross-section
ba
![Page 18: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/18.jpg)
Core LessonDiagonal Cross-section
Types of faces intersecteddetermines
types of edges on 2D figure
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Core Lesson
Intersects 2 arcs & 2 parallel lines
Diagonal Cross-section
![Page 20: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/20.jpg)
Core Lesson
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: rectangleDiagonal: half-moon
![Page 21: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/21.jpg)
How do you know what 2D shapes result from slicing through a cone?
What would this cone look like if we slice it diagonally?
![Page 22: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/22.jpg)
Let’s Review
Identify characteristics of the solid
Base
Lateral surface(face)
Edge
Apex
![Page 23: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/23.jpg)
Let’s Review Double-napped Cone
2 cones sharing 1 apex
Applications in algebraic
geometry & calculus
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Core Lesson
Vertical Cross-section
Vertical slice through apex always creates a triangle
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Core Lesson
Intersects 1 curved & 1 flat face:
parabola
Vertical Cross-section (cont.)
2 curved & 2 flat faces:
hyperbola
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Core Lesson
Intersecting curved lateral surface
Horizontal Cross-section
Geometrically similar to base:
circle
![Page 27: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/27.jpg)
A Common Misunderstanding
A diagonal cross-section creates a circle
Circles are only created by horizontal
cross-sections
![Page 28: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/28.jpg)
Core Lesson
Distance from center is not
constantActually an ellipse
Diagonal Cross-section
ba
![Page 29: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/29.jpg)
Core LessonDiagonal Cross-section
Types of faces intersecteddetermines
types of edges on 2D figure
![Page 30: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/30.jpg)
Core Lesson
Intersects 2 arcs & 2 parallel lines
Diagonal Cross-section
Hyperbola along the 2D plane
![Page 32: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/32.jpg)
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.
![Page 33: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/33.jpg)
How do you predict the 3D result of rotating a 2D figure?
What 3D shape would result from rotating this rectangle?
![Page 34: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/34.jpg)
Core Lesson
Axis bisects triangle
Rotating Triangle in 3D
Rotation creates a cone
![Page 35: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/35.jpg)
Core Lesson
Edges perpendicular to axis draw flat faces
Rectangle: Axis Bisecting
Edges parallel to axis draw curved surfaces
Rotation creates: cylinder
![Page 36: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/36.jpg)
Core Lesson
Edges perpendicular to axis draw flat faces
Rectangle: Axis Along Edge
Edges parallel to axis draw curved surfaces
Rotation creates: cylinder
![Page 37: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/37.jpg)
Core Lesson
Curved edges draw curved surfaces
Circle: Axis Bisecting
Rotation creates: sphere
![Page 38: 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.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cca5503460f949922f4/html5/thumbnails/38.jpg)
In this lesson you have learned how to predict the 3D
results of rotating simple figures by analyzing the
effects of rotations.