# Investigating cross-sections using GeoGebra

GeoGebra can be used to explore cross-sections. Here's one of the many cross-section interactives created by Edward Knote. This one allows the student to experiment with the various ways in which a plane and cube can intersect. We can ask students "what shapes can you make?" Some additional questions we might ask:

- Which n-gons can you make?
- Which regular n-gons?
- Which types of triangles?
- Which types of quadrilaterals?
- Why can't we make any curved shapes?

Another interesting activity which I've never seen done, is to match solids to animations of their cross-sections, as a plane passes through them. For example, if you pass a plane through a sphere, you'll see a very small circle appear, which then gets larger and larger, until it starts getting smaller again, returning to a very small circle once more, then disappearing altogether. For a cone, with the plane passing from the apex to the base, you would see a very small circle appear, which would get larger at a linear rate, then disappear once the plane has passed beyond the cone's base.

Here are a couple additional tidbits related to cross-sections: If you have a grid of cubes in 3D, and you cut them with a plane, there are several interesting cross-sections you can find, as seen below:

Also, if you have any tiling in 3-space, and you cut the tiling with a plane, you will get a tiling in 2-space:

It would be great to have an interactive for these in GeoGebra, but I wasn't able to find one as of today, March 1st, 2022. For more information on cutting 3-space tilings with planes, see this research paper.

I wonder how it would look to display the cross section of a plane and a 3D fractal, as the plane moves through it. For example, imagine a plane going through SierpiĆski's tetrahedron. The cross section of a plane and a grid of cubes, when the plane is at an angle, makes some interesting tilings. I wonder what would happen if we passed a plane through the Menger sponge at some angle.

There are two topics which will build on the idea of cross-sections: conic sections, which is a precalc topic, and volume of a solid of revolution, which is a calc 1 topic.