Lesson Plan - Get It!
- Have you ever watched those oddly satisfying videos of someone slicing into kinetic sand?
Check out a portion of the video below to see what I'm talking about!
Oddly Satisfying Relaxing ASMR Kinetic Sand Cutting Compilation 12 from stoakley hiroko:
In the video above, we see someone thinly slicing a half sphere, or hemisphere, repeatedly.
- Did you notice the shape of each slice?
If not, watch the video again and make note of the shape.
These slices are showing us the cross sections of a hemisphere. Simply put, a cross section is the shape we get as the result of 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:
Even doctors use cross sections when they examine a CT or MRI scan. This technology uses cross sections and is called cross-sectional imaging:
Let's explore what cross sections result from three-dimensional shapes with which we are familiar.
Start with a rectangular prism:
To find the cross sections, we imagine slicing through the prism using an intersecting plane.
Just like we saw in the video with kinetic sand, we can imagine slicing a rectangular prism with vertical slices (perpendicular to the base). When we do that, we 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 we decide to slice a rectangular prism horizontally (parallel to the base)?
We can imagine what would happen to our block of cheese, but be careful not to slice your fingers!
Slicing horizontally gives us cross sections with the same shape as the base. These are also rectangular cross sections.
- Are there other ways you can think of to slice a rectangular prism to get different cross sections?
- What happens if we start slicing the figure diagonally?
Notice that the cross sections shown above slice through four of the six faces of the rectangular prism.
Let's see what happens if we 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:
We see that the shape of the cross section depends on the number of faces we slice through.
To explore the different ways to slice through a rectangular prism, check out Sections of Rectangular Prisms (Cuboids) by Anthony Or for GeoGebra Institue of Hong Kong.
Now that we've discussed rectangular prisms, let's consider a different three-dimensional shape: pyramids.
- Have you ever wondered what the inside of the Egyptian pyramids looks like?
We can't take you to these pyramids to find out, but we 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 we 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 by Anthony Or for GeoGebra Institute of Hong Kong.
- Are you ready to start slicing through objects?
Move on to the Got It? section to check your understanding.