7. Geometry

Lesson

A cross-section is the 2D shape we get when cutting straight through a 3D object. For example, the orange below (let's pretend it's a perfect sphere) has been cut along the vertical plane (straight up and down). The cross-section of the orange, where we can see inside, is a circle.

Let's use the applet below to explore cross-sections that are **parallel** to the base of a prism and pyramid with the same polygon base.

- Drag the blue point in the 3D view up and down. What do you notice about the cross-section of the pyramid compared to the cross-section of the prism?
- Move the slider at the bottom to change the number of sides. Do your observations change when you increase the number of sides for the base?

Now let's explore some cross-sections that are **perpendicular** to the base of a prism and pyramid with the same polygon base.

- Drag the blue point in the 3D view to move the plane from left to right. What do you notice about the cross-section of the pyramid?
- What do you notice about the cross-section of the prism?
- Move the slider at the bottom to change the number of sides. Do your observations change when you increase the number of sides for the base?
- How do the vertical cross-sections compare to the horizontal cross-sections of the same type of solid?

Recall that prisms have rectangular sides, and the shape on the top and the base is the same. The name of the base shape gives the prism its name.

Prisms | |||||
---|---|---|---|---|---|

Triangular | Square | Rectangular | Pentagonal | Hexagonal | Octagonal |

Any cross-section taken parallel to the base in a prism is always the same shape and size as the base. In other words, we say that a prism has a uniform cross-section.

Recall that pyramids have triangular sides, and the shape of the base gives the prism its name.

Pyramids | |||||
---|---|---|---|---|---|

Triangular | Square | Rectangular | Pentagonal | Hexagonal | Octagonal |

In a pyramid, any cross-section taken parallel to the base is always the same shape but is smaller in size than the base.

From the applets, we can see that three-dimensional shapes have more than one cross-section and they may or not be the same shape. It all depends on which way we cut them!

We want to classify the following solid:

Does the shape have a uniform cross-section ?

Yes

ANo

BThe solid is a:

Triangular Pyramid

ARectangular Prism

BRectangular Pyramid

CTriangular Prism

D

Consider the solid in the adjacent figure.

If the solid is cut straight down below the dotted line, what cross-section results?

A pentagon

AA triangle

BA hexagon

CA square

DDoes the solid above have a uniform cross-section?

Yes

ANo

BWhat is the name of the solid?

A triangular prism

AA rectangular prism

BA hexagonal prism

CA pentagonal prism.

D

Which two of the objects below could have the following cross-section?

Triangular Pyramid

ACylinder

BPentagonal Pyramid

CPentagonal Prism

D

Describe the two-dimensional figures (cross sections) that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms, right rectangular pyramids, cones, cylinders, and spheres.