Skip to content

Volume of a Sphere, Cavalieri's Principle, Cups

?
Creation's quality: 0.0/5 (0 votes)
Evaluation of members on the printability, utility, level of detail, etc.
  • 429 views
  • 10 downloads

3D model description

####Making Sense of the Volume of a Sphere: Have a Cup of Math, Please.
#####Three different cups with a volume of approximately 2 fl oz.

The volume of a sphere of radius R is (4/3) πR^3. But why? One can certainly use a bit of calculus to find the volume of a sphere in secondary and college mathematics. In school mathematics, however, how can we make sense of the formula and, specifically, where does that (4/3) come from?

The present design allows one to explain the volume of a sphere using Cavalieri’s Principle. Specifically, it allows a semi-sphere of radius R to be aligned with a cylinder-minus-cone whose terminal radius and height are both R. When positioned appropriately (as shown in the figures), the cross-sections of the semi-sphere and the cylinder-minus-cone has the same area at any point of their height. This fact can be easily proved using the Pythagorean Theorem and simple comparison. Let’s start with the assumption that the volume of the cone of radius R and height R is (1/3) πR^3, which itself can be justified using Cavalieri’s Principle using a square pyramid as a reference (https://www.thingiverse.com/thing:3093547). Then, the cylinder-minus-cone has a volume of (2/3) πR^3, which is the same as that of the semi-sphere. Therefore, the volume of the whole sphere is (4/3)πR^3.

The cone can be taken from the cylinder in two ways: an upright cone and a tilted cone. Thus, there are two versions for the cylinder-minus-cone cup. In designing the cups, attention is paid to the inside of the cups so that liquid can be used to compare the volume (capacity) of both cups. The inner diameter is about 61mm, yielding a volume of approximately 2 fl oz.

####References
1. Cavalieri’s Principle. https://en.wikipedia.org/wiki/Cavalieri%27s_principle
2. https://www.britannica.com/science/Cavalieris-principle

  • 3D file format: STL
  • Publication date: 2020/06/22 at 09:43

Tags

Creator

Designs 218
Downloads 10.2k
Followers 88

STEAM educator, learning from and working with K-12 STEAM teachers to explore new ideas of teaching and engagement. I firmly believe ART is at the core of STEM learning or all human learning! I owe my ideas and designs to the hundreds of K-12 children and teachers and university professors I have had the pleasure of working with, in multiple disciplines-- math, science,engineering language arts, social studies, early childhood education and more! All mistakes, of course, are mine! There is no warranty or liability whatsoever implied or explicit behind the designs or ideas. They are all posted for their potential educational values.

When working with children, please strictly observe all safety and health procedures! Please refer to the NSTA safety guides: http://www.nsta.org/safety/.

LGBU Contact: LGBU@SIU.EDU

License

CC BY


Best sellers of the category Art


Add a comment


Would you like to support Cults?

You like Cults and you want to help us continue the adventure independently? Please note that we are a small team of 3 people, therefore it is very simple to support us to maintain the activity and create future developments. Here are 4 solutions accessible to all:

  • ADVERTISING: Disable your AdBlock banner blocker and click on our banner ads.

  • AFFILIATION: Make your purchases online by clicking on our affiliate links here Amazon, Gearbest or Aliexpress.

  • DONATE: If you want, you can make a donation via PayPal here.

  • WORD OF MOUTH: Invite your friends to come, discover the platform and the magnificent 3D files shared by the community!