Unit 12: Spatial Reasoning
Table of Contents
- 10.1 Solid Geometry
- Volumes and Surface Areas of Similar Cuboids
- 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
- 11 Nets of the Cube
- Cross Section? (1)
- 10.2 Representations of Three-Dimensional Figures
- 10.3 Formulas in Three Dimensions
- 10.4 Surface Areas of Prisms and Cylinders
- Unwrapping a Cylinder
- Unwrapping a Cylinder: REVAMPED!
- Right Triangular Prism!
- Adjustable Triangular Prism
- 10.5 Surface Area of Pyramids and Cones
- Square Pyramid: Underlying Anatomy
- oblique and right pyramid - Cavalieri
- Net of a Cone
- Curved Surface Area of Cones (Combined Version)
- 10.6 Volume of Prisms and Cylinders
- Mailbox? Backpack? (Composite Solid)
- Volume of a Rectangular Prism
- Washer Action!!! (1)
- Animation 169
- 10.7 Volume of Pyramids and Cones
- Volume of Pyramids
- Volume of Pyramids - Chinese Proof
- 10.8 Spheres
- Volume of Spheres
- Volume of Spheres with Proof
- A sphere in a cylinder.
- Surface Area of Spheres
- Sphere Peeling: Volume
- Cavalieri's principle
- Figure 6.7 Cavalieri's principle
- Cavalieri's Principle
Volumes and Surface Areas of Similar Cuboids
Unwrapping a Cylinder
Square Pyramid: Underlying Anatomy
Interact with this [b][color=#38761d]square pyramid[/color][/b] for a minute or two. [br][br]You can alter the base and height of this [color=#38761d][b]square pyramid[/b][/color] by dragging the 2 white points. [br][br]How would you describe the relationship between a square pyramid's [b]LATERAL HEIGHT[/b] and its [b]TRUE HEIGHT? [br][br][color=#1e84cc]To explore this resource in Augmented Reality, see the directions below the applet. [/color][/b]
Suppose each edge of the square base measures 10 cm. If this square pyramid's lateral height measures 13 cm, what would its true height be? [br][br]What would the pyramid's [b]lateral area[/b] be? What would its [b]total surface area[/b] be? [br][br]What would its [b]volume[/b] be?
TO EXPLORE IN AUGMENTED REALITY:
1) Open GeoGebra 3D app on your device. [br][br]2) Press the 3 horizontal bars (upper left corner). Select OPEN. [br][br]3) In the SEARCH TAB that appears, type [b]V9cAn76T[/b][br] Note this string of characters can be found in the URL here. [br] Be sure to either copy & paste this code or type it just the way you see it here. [br][br]4) The slider named [b]n [/b]does the entire animation. Slide this completely to the right. [br] When in 3D view, you can move the white point (outside the pyramid) up or down to change its height. [br] You can also drag the large white points on the pyramid's base to adjust the base size.
Mailbox? Backpack? (Composite Solid)
Use the WHITE VERTICES to alter the size of the composite solid shown below. [br][br]Be sure to answer the questions that follow.
1.
What 2 solids make up (COMPOSE) this composite solid? Describe.
2.
Suppose the semicircle (above) has a radius of 6 inches. [br]The distance between the [b][color=#ff7700]2 orange points[/color][/b] is 10 inches. [br]The distance between the [b][color=#ff7700]bottom orange point [/color][/b]and [b]bottom white point[/b] is 15 inches. [br][br]If this is so, describe how you would find the volume of this composite solid. [br]As you describe, find its volume along the way.
3.
The base of this composite solid ("bottom rectangle" is not shown.) Yet suppose, for this question, this rectangle is there. How would you find the total (outside) surface area of this composite solid? (Assume the same dimensions given in question (2).) . Find the total surface area of this composite solid within your description.
Volume of Pyramids
Volume of Spheres
The figure shows a hemisphere of radius [i]r[/i] and a cylinder of base radius and height [i]r[/i] with an inverted cone of the same height and base radius removed. Drag the red point to see the cross-sections of the two solids at a height [i]h[/i].[br](a) Express [i]x[/i] and [i]y[/i] in terms of [i]r[/i] and [i]h[/i].[br](b) Are the cross-sections equal in area?[br](c) Hence show that the volume of the sphere of radius [i]r[/i] is 4/3 π [i]r[/i]³.