## Volume of Spheres with GeoGebra

In the post “Volume of Pyramids with GeoGebra” some elementary proofs of the formula for the volume of a pyramid illustrated with GeoGebra are introduced. In this post, some methods of deriving the formula for the volume of spheres would be presented with the use of GeoGebra.

Eves (1990) presents a proof of the formula as follows. In the figure,
(Click the figure for the GeoGebra applet)
the area of the cross-section of the hemisphere

the area of the cross-section of the solid at the RHS

Hence the areas of the cross-sections of the two solids at the same height are always equal. By Cavalieri’s principle the volumes of two solids are equal. Hence the volume of the sphere

Ancient Chinese mathematicians approached the problem in a different way. In the classic problem book the Nine Chapters on the Mathematical Art, the volume of a sphere was incorrectly given as π²d³/16 (d = diameter). Chinese mathematician LIU Hui (劉徽, fl. 3rd century) explained the error as follows. It is known that the ratio of the area of a circle inscribed in a square to the area of the square is π/4. So the ratio of the volume of a cylinder inscribed in a cube to the volume of the cube is also π/4, because each cross-section is a circle inscribed in the square and the ratio of their areas is always π/4.

(Click the figure for the GeoGebra applet.)

Now consider also the sphere inscribed in the cube. If we assume, incorrectly, that the ratio of the volume of the sphere to the volume of the cylinder is also π/4, then we would get the volume of the sphere is π²d³/16.

LIU Hui pointed out that the object whose volume is 4/π times that of the sphere should be obtained by replacing each of its cross-section by the circumscribing square (see the figure). He called this object a “double box-lid” (Mouhefanggai 蓋合方蓋). He also pointed out that this object is the intersection of two orthogonal cylinder of the same radius (see the figure).

(Click the figure for the GeoGebra applet.)

LIU Hui was unable to find the volume of the double box-lid. The problem was only solved two centuries later by ZU Gengzhi (祖暅之), the son of the famous mathematician ZU Chongzhi (祖沖之). As shown in the figure below, ZU Gengzhi showed that the volume of the difference between an octant of the double box-lid and the cube containing it is equal to the volume of a pyramid whose volume is one-third of that of the cube. Hence the volume of the double box-lid, and also the volume of the sphere, were found.

(Click the figure for the GeoGebra applet.)

If we try to look at the area of the cross-section of the octant of the double box-lid directly, we could show that its volume is two-third of that of the cube, as shown in the following figures.

(Click the figure for the GeoGebra applet.)

Final Remark

It is hoped that the GeoGebra applets presented in these two posts on volumes of pyramids and spheres could help teachers to bridge the “logical gaps” (Tzanakis and Arcavi 2000 p.204-207) in the introduction of these formulas to lower form students without proofs. These materials are also valuable to provide problems to motivate and engage students in the teaching and learning of the formulas.

References

Eves, H. 1990. An introduction to the history of mathematics (Sixth Ed.). Saunders College Publishing

Tzanakis, C., Arcavi, A. 2000. ‘Integrating history of mathematics in the classroom: an analytic survey’, in J. Fauvel and J. van Maanen (eds.), History in Mathematics Education, Kluwer Academic Publishers, 201 – 240

Wagner, D . B. (1978) ‘Liu Hui and Tsu Keng-chih on the volume of a sphere’, Chinese Science Vol.3, 59-79

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## Followers Feature on GeoGebra Profile Pages

We are pleased to announce another great feature on your GeoGebra profile page! You can now ‘Follow’ your favorite people in the GeoGebra community and get updates whenever they publish new materials.

The new tab âFollowing’ lists all your Followers, as well as all people you are following, giving you easy access to their profile pages and public materials. As soon as other users start following you, you will receive a cool âFollowers badge’ on your profile page as well, counting the number of your Followers.

Please help us to spread the news, simply by signing in to your GeoGebra account and going to tab âFollowing‘ on your profile page. There, you may search for other users you think create great materials in the GeoGebra community and ‘Follow’ them to keep up-to-date with their newest contributions!

More new features coming soon! Share your ideas and feedback on our Help page.

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## GeoGebra Exam App

The new GeoGebra Exam AppÂ allows you and your students to use the power of GeoGebra during exams alongside paper and pencil (ie like a graphing calculator) while restricting their access to the Internet and other software on their computers that should not be used during the exam.

The GeoGebra Exam App …

• can be easily accessed viaÂ www.geogebra.org/exam
• runs in your browser (e.g. Chrome, Firefox, Internet Explorer 11, Safari), so no additional installations are required
• can be customized in case your students should not have access to the full power of GeoGebra, eg you can turn off CAS or 3D features for exams
• runs in fullscreen mode so students can’t use other programs or files during an exam

If a student leaves the GeoGebra Exam window (eg by opening another browser tab), the top header of GeoGebra Exam turns red immediately. An information button shows the Exam Log with detailed information when that happened and for how long.

For more details on how to best use the GeoGebra Exam App during tests and exams, please see our GeoGebra Exam Tutorial. We plan to include the Exam Mode in our offline apps (Chrome, tablet, phone) as well soon. For now, please give it a try and let us know any ideas for improvements at support@geogebra.org, thanks!

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## GeoGebra Exam Mode

The new GeoGebra Exam Mode allows you and your students to use the power of GeoGebra during exams alongside paper and pencil (ie like a graphing calculator) while restricting their access to the Internet and other software on their computers that should not be used during the exam.

The GeoGebra Exam Mode …

• can be easily accessed at www.geogebra.org/exam
• runs in your browser (e.g. Chrome, Firefox, Internet Explorer 11, Safari), so no additional installations are required
• can be customized in case your students should not have access to the full power of GeoGebra, eg you can turn off CAS or 3D features for exams
• runs in fullscreen mode so students can’t use other programs or files during an exam

If a student leaves the GeoGebra Exam window (eg by opening another browser tab), the top header of GeoGebra Exam turns red immediately. An information button shows the Exam Log with detailed information when that happened and for how long.

For more details on how to best use the GeoGebra Exam Mode during tests and exams, please see our GeoGebra Exam Tutorial. We plan to include the Exam Mode in our offline apps (Chrome, tablet, phone) as well soon. For now, please give it a try and let us know any ideas for improvements at support@geogebra.org, thanks!

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## GeoGebra Graphing Calculator Released

We’re pleased to announce the first public release of our GeoGebra Graphing Calculator! It’s for Android phones and tablets right now, iPhone and Windows will follow, watch this space!

For the first release it has the graphics and algebra views but we will be adding lots more features over the coming months. We have prepared the foundations so everything necessary is already working under-the-hood (for example the CAS engine, and you can create – but not yet display – 3D objects).

There are also some extra-cool goodies to enjoy:

• brand-new native equation editor
• long-tap on an object to change its properties
• drag sliders and animate them in the Algebra View
• freehand drawings and shape recognition
• everything is running natively (LaTeX, CAS, graphics drawing etc) so it is super-slick to use
• search, open and save from/to GeoGebraTube
• find out more in our tutorial Graphing Calculator Tutorial

You can download it in the Play Store now for Android phones and tablets – please let us know your ideas for improvements with the “Send Feedback” option or in our Mobile Apps forum.

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## GeoGebra Groups – Collaboration for Everyone

The GeoGebra community is all about creating and sharing great materials for learning math & science. Thanks to you, we now have more than 300,000 free and interactive worksheets and books.

But wouldn’t it be nice to also hear back from your students or colleagues who you shared your materials with? We think so too! That’s why we have created GeoGebra Groups:

• Supercool and easy collaboration between students and teachers, colleagues, and really just everyone nice out there!
• Invite others using your Group Code
• Post texts, images, videos, pdfs and GeoGebra worksheets to your Group
• Discuss your ideas with everyone or individual Group members using comments
• Find out how to do all that and more in our GeoGebra Groups Tutorial

Try out our Groups, start to collaborate in a new way, and let us know what you think!

Posts

Feedback

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## GeoGebra for Phones Beta

We’re pleased to announce the first public beta-release of GeoGebra for Android Phones. We’ve already started on an iPhone app, watch this space!

For the first release it will have just the Graphics View and Algebra View but we will be adding lots more features over the coming months. We have prepared the foundations already so everything necessary is already working under-the-hood (for example the CAS engine, and you can create – but not yet display – 3D objects).

There are also some extra-cool phone-only goodies to enjoy:

• brand-new native equation editor
• shake your phone to update random numbers (coming soon!)
• freehand drawings and shape recognition
• long-tap on an object to change its properties
• drag sliders and animate them in the Algebra View
• everything is running natively (LaTeX, CAS, graphics drawing etc) so it is super-slick to use

You can download it in the Play Store now (for Android phones and tablets!) – please let us know what you think with the “Send Feedback” option.

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## My World My Math!

Here’s a few things we really like in our world at GeoGebra:

1. Our community… yes, YOU 🙂
2. Our partners, and, last but not least …
3. Mathematics!

So we got together with a few amazing members of our community, one really awesome partner, and then turned the cameras ON. With a little fancy footwork, cutting, pasting… Voila!

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# GeoGebra Summer of Code 2015

This summer, GeoGebra was again part of Google Summer of Code (GSoC) supporting young and talented open source coders. We had a very strong field of 64 students applying and were able to choose 4 of them to work with us during this summer. There were also some impressive contributions submitted by many of the other candidates which have already been incorporated into GeoGebra.

## Yi-Chun (Jacky) Kuo: Native iOS App

Jacky

Our current iPad app is based on a WebView so we were interested to see if Jacky could make a prototype of a native app using the quite-new RoboVM cross-compiler to convert GeoGebra’s Java code into Objective C. As part of his application he managed to get the JLaTeXMath formula engine working which was very impressive in its own right. For the first half of his project he got the full GeoGebra kernel compiled and running and for the second half he worked on drawing objects and user-interaction.

There is still a lot to do before we can make a release (for example implementing the toolbar and equation editor) but Jacky will stay on with GeoGebra after GSoC to help!

GeoGebra running in Xcode

## Georgios Ouzounis: Handwritten Equation Recognition

Georgios

Handwritten equation recognized!

Georgios made a nice web app as part of his application to collect example handwritten equations from phones and tablets and store them in a database. This was important as it allowed him to use that data when testing and training his algorithms. The three parts to his project were:

• splitting the equation into its symbols
• recognising the symbols
• converting that back into GeoGebra syntax

and by the end of his project it was successfully recognizing indices, fractions and square-roots. At the moment the code is running server-side and is a little slow so the challenge for the future will be to get it running faster and client-side. Open Source libraries used: OpenCV

Polygon Intersections in 2D and 3D!

As part of Thilina’s application, he adapted the Clipper library to compile with Java 6 and with GWT and incorporated it into GeoGebra, which he then used as part of his main GSOC project to improve the polygon operations within GeoGebra.

He has also done a lot of work on improving the Intersect command to allow the intersection of many more object types to be calculated dynamically.

## Shamshad Alam: Implicit Curves Project

cos(x y) = k

x^4 + y^4 + z^4 = 2

Shamshad also made some nice experiments with the Clipper library as part of his application

The main focus of his GSoC project was to implement plotting of implicit curves and surfaces, namely:

• plotting of non-polynomial implicit curves (for example you can try sin(x) – sin(y) = 1 in GeoGebra Beta)
• plotting of implicit surfaces (eg x⁴ + y⁴ + z⁴ = 2)
• extending commands for rotation, translation and other transformations to work with these object types

Thanks to all our GSoC students for a great job done during this summer!

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## New Equation Editor and Keyboard

We’ve made some nice changes to our Chrome, Web and Tablet apps recently that many of you have been asking for. The first exciting feature is that the Input Bar in the Algebra View has expanded into a fully-fledged Equation Editor, for example:

and it also works fully with GeoGebra commands:

It’s now also much easier to add and edit matrices, parametric curves and piecewise-defined functions:

We’ve also added a touch-friendly on-screen keyboard that supports many languages and works nicely with our new equation editor:

You can try these cool new features in any of our web and tablet apps!

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