# Teaching and Learning Solid Geometry

Some students struggle understanding and visualizing solid geometry. Models can help. Although low-cost models exist for basic shapes and more expensive models are available for some complex shapes, the Thingiverse provides some 3D-printable objects for only the cost of printing. Plus, 3D-printed objects have a motivating cool-factor, which is Gagne's first--and often overlooked--event of instruction (i.e., Gain Attention and Interest).

Tad Novak (tadman1996) and bbennington both posted two good sets of basic shapes:

Kyle Martin (KyleMiles) also posted an ellipsoid and paraboloid:

Are you aware of other objects that could be useful for teaching and learning about solid geometry?

Any other teaching approaches for this topic (e.g., Kinesthetic Math or Tactile Learning) that might benefit from 3D-printed learning objects?

Elementary Geometric Shape Set
Shapes
Math Shapes!

BlocksCAD is another free online software that can help students understand shapes! Check us out at: https://www.blockscad3d.com/editor/

I like the free program K3DSurf, which generates 3D meshes by entering parametric equations There is a long drop-down list of example shapes. Depending on equation a non-solid surface is often generated, but can be extruded into a 3D-printable shape with something like meshmixer or blender.

Thanks for recommending K3DSurf. It reminded me of the challenges of teaching, graphing, visualizing, and understanding 3D equations. Do you think K3DSurf could be used to create a model illustrating a common 3D graph, such as a saddle point?

If so, and you're experienced with K3DSurf and 3D equations, could you create and post a few models? I think it could generate some useful work in this area.

Bill Owens (owens) posted some unusual polyhedrons and a good explanation of cuboctahedron, octahemioctahedron, and cubohemioctahedron for the curious. His work could be an great enrichment activity for those genius students who love geometry.

Stewart B4,3 and B4,4 polyhedra
by owens

K3Dsurf can be used to make saddle points/paraboloids, hyperboloids, and other parametric curvy shapes....But I'm going to take the easy way out and just point you to Elizabeth Denne's designs :)

https://www.thingiverse.com/dennedesigns/designs/page:1

If there's another specific shape you are interested in, let me know and I can probably whip it up.

Tomas de-Camino-Beck (Automata) recently posted "Disk method of approximating a sphere", great model set for understanding the disk method for Volume of Revolutions. His set includes six models estimating the volume of a half sphere. Models start with four disks roughly estimating the volume, and they progress to a model with about 50 disks that more accurately estimates the volume. Tomas suggests submerging them in water to determine their volume, and he includes a final half-sphere model for a baseline.

Disk method of approximating a sphere

Thanks for the excellent lead into Elizabeth Denne's work. My favorites are:

because they help students visualize Volume of Revolutions.

Sphere: 10 disks
Volume by 16 French Fries

Daniel Meldrim (xeno108) recently posted a great visual comparing the disc and shell methods for volume of revolutions. When I taught Calculus, students often asked which method was better. Although one method might be more straightforward than the other depending on the situation, either method works. Daniel's "Calculus tools: Disc and Shell Method visual aide" helps illustrate and compare the two methods.

Calculus tools: Disc and Shell Method visual aid
by xeno108