Integrated math 2
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
- 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.
|Term||Meaning in polyhedra||With figures with curved surfaces, we also mean:|
|Face||A flat surface||A continuous surface|
|Edge||A line segment where 2 faces meet||A line segment or curve where 2 surfaces meet|
|Vertex||A 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
only applies to polyhedra, so we'd use the meanings in polyhedra. Words adapt and gain new meanings based on need.
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 plane||Sample figure and planes||Cross-sections|
|Parallel to the base|
|Perpendicular to the base|
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 face||A straight edge|
|A curved face||A curved edge (usually)*|
|Parallel faces||Parallel edges|
|An edge||A vertex|
|A vertex||A 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.
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?
Want to join the conversation?
- hello homosapiens(10 votes)
- So what are you a Neanderthal?
I'm an alien, anyway, hello human I'm going to invade earth!
Sorry, I had to. :)(1 vote)
- Why is it so complicated(7 votes)
- What do you use apexes for?(2 votes)
- The apex is used to find the height of the 3D pyramid or cone. In a skewed cone or pyramid, the apex does not have to "above" the figure, so the height would be from the apex to the plane of the base.(9 votes)
- What does some of this stuff mean? Because i understand it but its kinda hard for me because all of these formulas and these vertexes and apex Has me kinda confused.(5 votes)
- What's a polyhedron?(3 votes)
- A polyhedron is a solid figure where every surface is a polygon. Prisms and pyramids are examples of polyhedra.(5 votes)
- What's parallel plane?(2 votes)
- whats an apex(2 votes)