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## High school geometry

### Course: High school geometry>Unit 9

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.
A triangular prism.
A rectangular prism.
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.
A prism-like figure where the two bases are quarter circles. It has two rectangular sides and one curved side.
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 rectangular prism and an oblique rectangular prism. The oblique prism has the same dimensions, but it leans to the right at the top. Both of the figures are cut into congruent cross sections.
• A right prism has its top face directly above its bottom face. The translation vector is perpendicular to the bases.
A trapezoidal prism
• An oblique prism has a non-perpendicular translation vector.
An oblique triangular prism that leans to the right at the top. A height segment of the prism does not coincide with a lateral edge of the prism.

### 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.
A rectangular pyramid.
An oblique rectangular pyramid that leans to the left at the top. A height segment of the prism does not coincide with a lateral edge of the prism.
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.
A cone.
An oblique cone that leans to the right at the top. A height segment of the prism does not coincide with a lateral edge of the prism.
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.
An oblique triangular pyramid and a triangular pyramid. The oblique pyramid has the same dimensions, but it leans to the left at the top. Both of the figures are cut into cross sections. At any given height, the cross-section of one pyramid is congruent to the cross-section of the other pyramid
• A right pyramid has its apex directly above the center of the base.
A right trapezoidal pyramid.
• An oblique pyramid has its apex anywhere else.
An oblique rectangular pyramid that leans to the left at the top. A height segment of the prism does not coincide with a lateral edge of the prism.

### 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.
A sphere.

## 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 right rectangular prism. One of the six flat surfaces is highlighted.
A continuous surface
A cone. The curved surface of the cone is highlighted.
EdgeA line segment where 2 faces meet
A rectangular pyramid. The outsides are highlighted.
A line segment or curve where 2 surfaces meet
A cylinder. The outsides of the circular bases are highlighted.
VertexA point where 2 or more edges meet
A triangular prism where its vertices are highlighted.
The point opposite to and farthest from the base of the figure (also called an apex)
A cone where the tip of it is highlighted.
This is a good reminder that the definition of a word depends on context. For example, Euler's formula start text, v, e, r, t, i, c, e, s, end text, plus, start text, f, a, c, e, s, end text, minus, start text, e, d, g, e, s, end text, equals, 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
A cone. There is a rectangle that cuts through the cone parallel to the base. The rectangle represent a cross section cut.
A circle.
Perpendicular to the base
A cone with a rectangle through it that represents the cross-section. The rectangle is at an angle so that it is perpendicular to an base of the cone. The rectangle moves through the entire cone showing every possible cross-section.
A cone. There is a rectangle that cuts through the cone from the base to the curved side. The rectangle represent a cross section cut.
A triangle.
A cross section of a cone that looks like a triangle with a curved top. This shape is called a hyperbola.
Diagonal
A cone with a rectangle moving from the base to the apex to show the cross sections. The rectangle is diagonal to the cone's base, so it makes varying sizes of ellipses, from largest to smallest. When the rectangle crosses the base, it makes a shape with one curved side and one straight side.
A cone with a rectangle moving from the base to the apex to show the cross sections. The rectangle is diagonal to the cone's base, so it makes varying sizes of ellipses, from largest to smallest. When the rectangle crosses the base, it makes a shape with one curved side and one straight side.
An ellipse.
An ellipse but the right side is vertically cut.
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
• Current
The following set of cross-sections are the intersections of equally-spaced planes parallel with the base of a solid.
A series of triangles growing in size from left to right.
Which kinds of solids could the cross-sections be from?