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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 -. - PowerPoint PPT Presentation
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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 -
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?
Let’s Review
Parts of a solid
Face, edge, and vertex
(vertices)
Apex
Core Lesson
Effect of slicing plane
Where plane intersects faces, edges of 2D figure results
Core Lesson
Identify characteristics of the solid
SquareTriangles
Core Lesson
Vertical, through apex
=triangle
Vertical slice through apex
Core LessonHorizontal Cross-section
Slices parallel to the base will always be similar to the base
A Common Misunderstanding
A plane can slice through a solid in
any directioncross-sections are always horizontal or vertical
Core Lesson Number of intersected faces
=number of edges
4 faces/edges5 faces/edges
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
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?
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.
Let’s Review
Identify characteristics of the solid
BasesLateral surface(face)
Edges
Core Lesson
Distance from center is constant.Therefore it’s a
circle
Horizontal Cross-section
rr
Slices parallel to base are congruent
to base
Core Lesson
Vertical Cross-section
Vertical slice always creates a parallelogram
A Common Misunderstanding
A diagonal cross-section creates a circle
Circles are only created by horizontal
cross-sections
Core Lesson
Distance from center is not
constantActually an ellipse
Diagonal Cross-section
ba
Core LessonDiagonal Cross-section
Types of faces intersecteddetermines
types of edges on 2D figure
Core Lesson
Intersects 2 arcs & 2 parallel lines
Diagonal Cross-section
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
How do you know what 2D shapes result from slicing through a cone?
What would this cone look like if we slice it diagonally?
Let’s Review
Identify characteristics of the solid
Base
Lateral surface(face)
Edge
Apex
Let’s Review Double-napped Cone
2 cones sharing 1 apex Applications in
algebraic geometry &
calculus
Core Lesson
Vertical Cross-section
Vertical slice through apex always creates a triangle
Core Lesson
Intersects 1 curved & 1 flat face:
parabola
Vertical Cross-section (cont.)
2 curved & 2 flat faces:
hyperbola
Core Lesson
Intersecting curved lateral surface
Horizontal Cross-section
Geometrically similar to base:
circle
A Common Misunderstanding
A diagonal cross-section creates a circle
Circles are only created by horizontal
cross-sections
Core Lesson
Distance from center is not
constantActually an ellipse
Diagonal Cross-section
ba
Core LessonDiagonal Cross-section
Types of faces intersecteddetermines
types of edges on 2D figure
Core Lesson
Intersects 2 arcs & 2 parallel lines
Diagonal Cross-section
Hyperbola along the 2D plane
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.
How do you predict the 3D result of rotating a 2D figure?
What 3D shape would result from rotating this rectangle?
Core Lesson
Axis bisects triangle
Rotating Triangle in 3D
Rotation creates a cone
Core Lesson
Edges perpendicular to axis draw flat faces
Rectangle: Axis Bisecting
Edges parallel to axis draw curved surfaces
Rotation creates: cylinder
Core Lesson
Edges perpendicular to axis draw flat faces
Rectangle: Axis Along Edge
Edges parallel to axis draw curved surfaces
Rotation creates: cylinder
Core Lesson
Curved edges draw curved surfaces
Circle: Axis Bisecting
Rotation creates: sphere
In this lesson you have learned how to predict the 3D
results of rotating simple figures by analyzing the
effects of rotations.
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