A cross section is the 2D shape we get when cutting straight through a 3D object. For example, the orange (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.

Ideas

Parallel cross sections of prisms and pyramids
Perpendicular cross sections of prisms and pyramids

Parallel cross sections of prisms and pyramids

Exploration

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?

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By using the applet above, we can notice that the parallel cross section of a prism results in the same shape and size of its base. The parallel cross section of a pyramid results in the same shape of its base but smaller.

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.

Table of different kinds of prisms containing triangular, square, rectangular, pentagonal, hexagonal, and octagonal prisms.

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.

Table of kinds of pyramids containing triangular, square, rectangular, pentagonal, hexagonal, and octagonal pyramids.

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

Examples

Example 1

We want to classify the following solid:

Does the shape have a uniform cross section ?

Worked Solution

Create a strategy

We need to classify the 2 faces that are identical to each other and parallel. If we cut through the solid parallel of the 2 faces, do we keep getting the same cross section?

Apply the idea

The 2 identical faces of the solid figure is triangle, and when we cut through the solid parallel we keep on getting the exact same shape and size of the faces. So the shape does have a uniform cross section.

The solid is a:

Triangular pyramid

Rectangular prism

Rectangular pyramid

Triangular prism

Worked Solution

Create a strategy

Determine if the solid is a prism or pyramid and determine the shape of its base.

Apply the idea

The solid is a prism because it has a uniform cross section and the shape of its base is triangle, so the solid is a triangular prism.

The correct option is D: Triangular prism.

Example 2

Consider the solid in the adjacent figure.

A hexagonal prism. A horizontal dotted line is drawn above the prism.

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

A pentagon

A triangle

A hexagon

A square

Worked Solution

Create a strategy

If the solid is cut straight down below the dotted line the cross section results the same as the front face.

Apply the idea

The front face of the solid is a hexagon, this means that the cross section is also a hexagon because it will be the same shape as its face.

The correct option is C: A hexagon.

Does the solid above have a uniform cross section?

Worked Solution

Create a strategy

We need to classify the 2 faces that are identical to each other and parallel. If we cut through the solid parallel of the 2 faces, do we keep getting the same cross section?

Apply the idea

The 2 identical faces of the solid figure is hexagon, and when we cut through the solid parallel we keep on getting the exact same shape and size of the faces. So using the parallel cutting, the shape does have a uniform cross section.

What is the name of the solid?

A triangular prism

A rectangular prism

A hexagonal prism

A pentagonal prism

Worked Solution

Create a strategy

We can determine the name of the prism according to the name of its uniform cross section.

Apply the idea

The name of the uniform cross section in the solid is hexagon, so the given solid is a hexagonal prism.

The correct option is C: A hexagonal prism.

Example 3

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

Triangular pyramid

Cylinder

Pentagonal pyramid

Pentagonal prism

Worked Solution

Create a strategy

Any object with a pentagonal base should have a pentagonal cross section.

Apply the idea

The correct options are C and D because they have a pentagonal base.

Idea summary

A prism has a uniform cross section which meant that any cross section taken parallel to the base in a prism is always the same shape and size as the base.

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

Perpendicular cross sections of prisms and pyramids

Exploration

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?

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By using the applet above, we can notice that a perpendicular cross section of a pyramid is a polygon. And a perpendicular cross section of a prism is a rectangle.

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.

Examples

Example 4

Suppose you are cutting each of the following solids perpendicular to their bases. Which solid does NOT have cross sections that are always rectangles?

Worked Solution

Create a strategy

The perpendicular cross section of a prism is a rectangle while that of a pyramid is a polygon.

Apply the idea

Option A is a square prism.

Option B is a rectangular prism.

Option C is a rectangular pyramid.

The solid with cross section that is not always a rectangle is a pyramid, which is option C.

Idea summary

A prism has a perpendicular cross section that is always a rectangle.

A pyramid has perpendicular cross sections that can be a triangle or a polygon.

Outcomes

7.G.A.3

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

7.04 Cross sections of prisms and pyramids

Introduction

Ideas

Parallel cross sections of prisms and pyramids

Exploration

Examples

Example 1

Example 2

Example 3

Perpendicular cross sections of prisms and pyramids

Exploration

Examples

Example 4

Outcomes

7.G.A.3

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