*Contributor: Michelle Haver. Lesson ID: 13627*

Three-dimensional shapes make up the world around you. Explore what happens if you slice through those objects and see what shapes are left behind!

categories

subject

Math

learning style

Visual

personality style

Beaver

Grade Level

Middle School (6-8)

Lesson Type

Quick Query

- Have you ever watched someone slicing into kinetic sand?

Check out a portion of the video below to see what it's all about!

- Pretty cool, right?

In the video above, someone thinly slices a half sphere or hemisphere repeatedly .

- Did you notice the shape of each slice?

If not, watch the video again and note the shape.

These slices are showing the **cross-sections** of a hemisphere. Simply put, a cross-section is the shape you get from slicing through an object.

You have likely seen cross-sections in real life every day!

For example, a slice of bread gives the cross-section of the whole loaf.

Doctors use cross-sections when examining a CT or MRI scan. This technology uses cross-sections and is called *cross-sectional imaging*.

Explore what cross-sections result from three-dimensional shapes with which you are familiar.

Start with a rectangular prism.

To find the cross-sections, imagine slicing through the prism using an intersecting plane.

Just like in the video with kinetic sand, you can imagine slicing a rectangular prism with *vertical* slices (perpendicular to the base). When you do that, you get cross -sections with the same shape as the faces on the left and right sides.

These cross-sections are **rectangles**.

You may be thinking that the blue shape in the middle doesn't look like a rectangle — it looks more like a diamond shape from this angle! It may be helpful to picture the shape sitting in front of you.

For example, think about how you could slice a block of cheese to see what cross-sections you get from a rectangular prism!

- What happens if you decide to slice a rectangular prism
*horizontally*(parallel to the base)?

You can imagine what would happen to the block of cheese, but be careful not to slice your fingers!

Slicing horizontally gives cross-sections with the same shape as the base. These are also **rectangular cross-sections**.

- Are there other ways to slice a rectangular prism to get different cross-sections?
- What happens if you start slicing the figure diagonally?

Notice that the cross sections shown above slice through four of the six faces of the rectangular prism.

See what happens if you slice through different numbers of faces using *diagonal slices.*

Slicing through three faces diagonally results in a **triangular cross-section**.

Slicing through four faces diagonally results in a **quadrilateral cross-section**.

Slicing through five faces diagonally results in a **pentagonal cross-section**.

Slicing through six faces diagonally results in a **hexagonal cross-section**.

You see that the shape of the cross-section depends on the number of faces you slice through.

To explore the different ways to slice through a rectangular prism, check out Sections of Rectangular Prisms (Cuboids).

Now that you've discussed rectangular prisms, consider a different three-dimensional shape: pyramids.

- Have you ever wondered what the inside of the Egyptian pyramids looks like?

This lesson can't take you to these pyramids to find out, but you can explore what it would be like to slice through the pyramids by considering cross-sections!

Imagine you have a pyramid with a rectangular base sitting in front of you. Think of the different ways you could slice through the pyramid.

- What cross-sections can you make as a result of slicing a
**right rectangular pyramid**?

Remember to consider *vertical slices* (perpendicular to the base), *horizontal slices* (parallel to the base), and *diagonal slices*.

To explore the possible cross sections from taking slices of a right rectangular prism, check out Sections of Rectangular Pyramids.

- Are you ready to start slicing through objects?

Move to the *Got It?* section to check your understanding.