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
Apex
Face,
edge,
and vertex
(vertices)
Core Lesson
Effect of slicing plane
Where plane
intersects faces,
edges of 2D figure
results
Core Lesson
Identify characteristics of the solid
Triangles
Square
Core Lesson
Vertical slice through apex
Vertical,
through apex
=
triangle
Core Lesson
Horizontal 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
cross-sections are always
any direction
horizontal or vertical
Core Lesson
Number of
intersected faces
=
number of edges
4 faces/edges
5 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
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.
In this lesson, you will learn how
to visualize the 2D crosssections of cylinders by analyzing
if a plane intersects with straight
or curved surfaces.
Let’s Review
Identify characteristics of the solid
Edges
Lateral
surface
(face)
Bases
Core Lesson
r
r
Horizontal Cross-section
Distance from
center is constant.
Therefore it’s
a circle
Slices parallel to
base are
congruent to base
Core Lesson
Vertical Cross-section
Vertical slice
always creates a
parallelogram
A Common Misunderstanding
Circles are only
A diagonal cross-section
created by horizontal
creates a circle
cross-sections
Core Lesson
Diagonal Cross-section
Distance from
center is
not constant
Actually an ellipse
Core Lesson
Diagonal Cross-section
Types of faces
intersected
determines
types of edges on
2D figure
Core Lesson
Diagonal Cross-section
Intersects 2 arcs &
2 parallel lines
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: rectangle
Diagonal: 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
Apex
Edge
Lateral
surface
(face)
Base
Let’s Review
2 cones
sharing 1 apex
Double-napped Cone
Applications in
algebraic
geometry &
calculus
Core Lesson
Vertical Cross-section
Vertical slice
through apex
always creates a
triangle
Core Lesson
Vertical Cross-section (cont.)
Intersects 1 curved
& 1 flat face:
parabola
2 curved & 2 flat
faces:
hyperbola
Core Lesson
Horizontal Cross-section
Intersecting curved
lateral surface
Geometrically
similar to base:
circle
A Common Misunderstanding
Circles are only
A diagonal cross-section
created by horizontal
creates a circle
cross-sections
Core Lesson
Diagonal Cross-section
Distance from
center is
not constant
Actually an ellipse
Core Lesson
Diagonal Cross-section
Types of faces
intersected
determines
types of edges on
2D figure
Core Lesson
Diagonal Cross-section
Intersects 2 arcs &
2 parallel lines
Hyperbola along
the 2D plane
Core Lesson
Interactive
In this lesson, you have learned
how to visualize the 2D crosssections 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
Rotating Triangle in 3D
Axis bisects
triangle
Rotation creates
a cone
Core Lesson
Rectangle: Axis Bisecting
Edges perpendicular
to axis draw flat faces
Edges parallel to axis
draw curved surfaces
Rotation creates:
cylinder
Core Lesson
Rectangle: Axis Along Edge
Edges perpendicular
to axis draw flat faces
Edges parallel to axis
draw curved surfaces
Rotation creates:
cylinder
Core Lesson
Circle: Axis Bisecting
Curved edges draw
curved surfaces
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.