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## Integrated math 2

### Course: Integrated math 2>Unit 10

Lesson 1: 2D vs. 3D objects

# Solid geometry vocabulary

Learn the names of common solid figures, the parts of those solids, and how we describe cross-sections of them.

## Types of 3D solids

### Prisms and prism-like figures

A prism is a pair of congruent polygons in parallel planes and the collection of all the points between them.
We'll use prism-like figure to mean any figure that is like a prism, except that the base can be any 2D shape. The most common prism-like figure is a cylinder.
Another way to think of prisms and prism-like figures is that they are the collection of translations of the base. Every cross-section of a prism parallel to its base has the same area.
• A right prism has its top face directly above its bottom face. The translation vector is perpendicular to the bases.
• An oblique prism has a non-perpendicular translation vector.

### Pyramids and pyramid-like figures

A pyramid is a polygon, a vertex in a different plane, and the collection of all the points between them.
We'll use pyramid-like figure to mean any figure that is like a pyramid, except that its base can be any 2D shape. The most common pyramid-like figure is a cone.
Another way to think of pyramids and pyramid-like figures is that they are the collection of dilations of the base about the apex for all scale factors from $0$ to $1$.
• A right pyramid has its apex directly above the center of the base.
• An oblique pyramid has its apex anywhere else.

### Other common figures

A polyhedron is a solid figure where every surface is a polygon. Prisms and pyramids are examples of polyhedra.
A sphere is a solid figure where every point on the surface is the same distance from its center.

## Parts of 3D solids

There is a lot of useful vocabulary related to polyhedra, but not as much related vocabulary for the corresponding features of 3D objects with curved surfaces.
For the sake of communication, we're going to extend the vocabulary from polyhedra to other 3D figures as well.
TermMeaning in polyhedraWith figures with curved surfaces, we also mean:
FaceA flat surface
A continuous surface
EdgeA line segment where 2 faces meet
A line segment or curve where 2 surfaces meet
VertexA point where 2 or more edges meet
The point opposite to and farthest from the base of the figure (also called an apex)
This is a good reminder that the definition of a word depends on context. For example, Euler's formula $\text{vertices}+\text{faces}-\text{edges}=2$ only applies to polyhedra, so we'd use the meanings in polyhedra. Words adapt and gain new meanings based on need.

### Cross-sections

The intersection of a plane and a solid is a cross-section. So every cross-section is 2D figure that we could get by slicing through a 3D figure.
Orientation of the planeSample figure and planesCross-sections
Parallel to the base
Perpendicular to the base
Diagonal
We'll try to specify whether the plane is perpendicular or parallel to the figure's base (or neither) when we ask about a cross-section. In some textbooks, if they don't specify the orientation of the plane, they mean that the plane is parallel to the base. In other books, the plane could face any direction. So be sure to check which meaning your class is using for any classwork.
Slicing through (in the 3D shape)Creates (in the 2D cross-section)
A flat faceA straight edge
A curved faceA curved edge (usually)*
Parallel facesParallel edges
An edgeA vertex
A vertexA vertex
*There are a few exceptions where you can slice through a curved surface and create a straight edge. Here are the two most common exceptions:
• Slicing a right cylinder perpendicular to its base creates straight edges.
• Slicing a cone through its apex creates straight edges.
Problem 1.1
The following set of cross-sections are the intersections of equally-spaced planes parallel with the base of a solid.
Which kinds of solids could the cross-sections be from?