3D Figures Volume & Cross-Sections
Modified version of resources originally posted by Tim Brizinski.
Table of Contents
- Perimeter, Area, Volume
- Circumference = ? (Animation)
- Circumference = ? (Animation II)
- Animation 65
- Parallelogram: Area
- Building Parallelograms with Given Area: Quick Formative Assessment
- Triangle Area Action!!! (V2)
- Create a Triangle with Given Area: Quick Formative Assessment
- Circle Area (by Peeling)
- Area of a Circle (III)
- Area of a Circle (by Peeling)
- Area of a Circle II
- Volume of a Rectangular Prism
- Figure 6.7 Cavalieri's principle
- Cavalieri's Principle
- Mailbox? Backpack? (Composite Solid)
- Washer Action!!! (1)
- Trisecting the Cube into 3 Pyramids
- Volume of Pyramids
- More with volume
- Unwrapping a Cylinder: REVAMPED!
- A sphere in a cylinder.
- Square Pyramid: Underlying Anatomy
- Cone Anatomy
- Build Your Own Right Triangular Prism (V2)!
- Build Your Own Right Triangular Prism!
- Right Triangular Prism!
- Sphere Peeling: Volume
- Net of a Cone
- 2D to 3D and Cross Section
- Rotating 2D Graphs about Lines to Create 3D Surfaces of Revolution in GeoGebra AR
- Surface of Revolution: GeoGebra Augmented Reality Template
- Sphere + Plane = ?
- Sections of Rectangular Prisms (Cuboids)
- Sections of Triangular Prisms
- Sections of Cylinders
- Sections of Rectangular Pyramids
- Sections of Triangular Pyramids
- Sections of Cones
- Sections of Spheres
- Exploring Sections of Cubes
- Cross Section? (1)
- Other Geometry Resources
Circumference = ? (Animation)
[b][color=#1551b5]Let's try to find the circumference of a circle by wrapping diameters around it.[/color][/b][br][br][b][color=#c51414]Hit the "Start Animation!" button below and carefully observe what happens.[/color][/b][br][i][color=#0a971e](Be sure to answer the 2 questions that follow.)[/color][/i]
Unwrapping a Cylinder: REVAMPED!
A [b][color=#0000ff]HUGE THANK YOU[/color][/b] and [b][color=#0000ff]CREDIT ACKNOWLEDGEMENT[/color][/b] to both [url=https://www.geogebra.org/bernat_geogebra]Bernat[/url] and [url=https://www.geogebra.org/stevephelps]Steve Phelps[/url], who calculated the surface equations for the [b][color=#ff00ff]curved parts of the cylinder's lateral area[/color][/b] (during the unwrapping process). [br][br]This enhanced version (of [url=https://www.geogebra.org/m/nY92fgyN]this original version[/url]) would not have been possible without their insight. [br][br][b][color=#1e84cc]To explore this resource in GeoGebra Augmented Reality, see the directions below this applet. [/color][/b]
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to the MENU (horizontal bars) in the upper left corner. Select OPEN. [br] In the Search GeoGebra Resources input box, type [b]rCxXxFhE[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br]3) In the resource that uploads, zoom in/out if needed.[br] The [b]r[/b] slider adjusts the radius of the cylinder.[br] The [b]h [/b]slider adjusts the height of the cylinder.[br] Drag the slider named [b]i[/b] to wrap/unwrap.
Rotating 2D Graphs about Lines to Create 3D Surfaces of Revolution in GeoGebra AR
Students & Teachers with ARCore by Google Installed on Your Devices:
Here's how you can create a 3D surface by rotating the graph of ANY function about the xAxis within [url=https://play.google.com/store/apps/details?id=org.geogebra.android.g3d]GeoGebra's 3D Graphing Calculator app[/url]. (This screencast was recorded on an ACER Chrome Tab 10.) [br][br]For more information re: getting started with GeoGebra Augmented Reality, [url=https://www.geogebra.org/m/jhcywuqw]click here[/url].
Quick Demo
Now, press the [b]AR [/b]button (lower right) to explore this within GeoGebra Augmented Reality! [br][br]Once viewing in Augmented Reality, slide the slider slowly. You should see a surface of revolution form!
Here's what this surface looks like in GeoGebra Augmented Reality!
For further exploration:
[b]Note[/b]:[br][br]If interested, you can use this pre-made template [url=https://www.geogebra.org/m/qbxbcmqw#material/yt22anmh]that can be found here[/url]. [br]Follow the directions on this page to open in the [color=#1e84cc]GeoGebra 3D Graphing Calculator[/color] on your device. [br][br]1) Instead of f(x) = 3, try typing [math]f\left(x\right)=x,0\le x\le4[/math] (just the way you see it). Spin this graph about the [br] xAxis. What kind of solid do you get? [br][br]2) Same direction as in (1), but do this time for [math]f\left(x\right)=x,2\le x\le4[/math]. What do you get now? [br][br]3) Same direction as in (1), but do this time for [math]f\left(x\right)=0.5x^2,0\le x\le2[/math] . What does this look like? [br][br]4) What other kinds of 3D surfaces of revolution can we create this way? [br] Let your creativity take you places!