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 Square Triangles

Vertical, through apex = triangle Vertical slice through apex

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 any direction cross-sections are always 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

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 Bases Lateral surface (face) Edges

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

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 constant Actually an ellipse Diagonal Cross-section b a

Core Lesson Diagonal Cross-section Types of faces intersected determines types of edges on 2D figure

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: 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 Base Lateral surface (face) Edge Apex

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

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

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 constant Actually an ellipse Diagonal Cross-section b a

Core Lesson Diagonal Cross-section Types of faces intersected determines types of edges on 2D figure

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

Core Lesson Interactive

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

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

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

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.