Crosssections of Three-Dimensional Objects

CrossSections of Three-Dimensional Objects

The Lesson Activities will help you meet these educational goals:

• Content Knowledge—You will identify the shapes of two-dimensional cross sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects.
• Inquiry—You will make observations and draw conclusions.

• 21st Century Skills—You will carry out technology-assisted modeling.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

1. Cross Sections

In this task, you will use cross sectionflyer, an online tool, to investigate the cross sections of cones, cylinders, pyramids, and prisms by passing different planes through such objects.

For each part of this task, you willselect a three-dimensional object from the list of shapes in the bottom-left corner of the screen. A three-dimensional view of the objectappears on the left, and the two-dimensional cross section formed by the intersection of the object with thechosen plane appears on the right. The bottom-right controls allow you to rotate and move the plane of intersection.By varying the intersecting plane, you can change the resulting cross section.

You can also adjust the view of the three-dimensional object by clicking and dragging anywhere inside the object window.Use the coordinate X-Y-Zaxes to guide you through the views. You can verify the name of a cross section by using the Click to show identification of cross sectioncheckbox.

1. Select Cone in the list of shapes. Vary the angle between the intersecting plane and the vertical axis, and identify the shapes of the cross sections formed. Try to form as many different cross-sectional shapes as possible. You might find it helpful to start with a horizontal intersecting plane and gradually increase the angle of the plane until it is vertical. Be sure to observe the effect of moving the plane without rotating it.

1. In the table, describe the shape of the cross sectionformed when a particular plane passes through the cone.

Type your response here:

Description of Plane / Description ofCross Section
plane parallel to the circular base, not passing through the tip of the cone
plane parallel to the circular base, passing through the tip of the cone
plane not parallel to the base, not passing through the base, and making an angle with the horizontal that is less than that made by the slant height of the cone
plane not parallel to the base, passing through the base, and making an angle with the horizontal that is less than that made by the slant height of the cone
plane parallel to the slant height of the cone, passing through the tip of the cone
plane parallel to the slant height of the cone, not passing through the tip of the cone
plane making an angle with the horizontal that is greater than the angle made by the slant height, not passing through the tip of the cone
plane making an angle with the horizontal that is greater than that made by the slant height, passing through the tip of the cone
plane parallel to the vertical axis, not passing through the tip of the cone
plane parallel to the vertical axis, passing through the tip of the cone
other planes through the tip of the cone

1. Which cross-sectional shapes do you find the most surprising? Which shapes do you find the least surprising? Explain why.

Type your response here:

1. Select Cylinder from the list of shapes. Move the intersecting plane, and vary the angle between the intersecting plane and the vertical axis to identify the shapes of different types of cross sections. Try to form as many different cross-sectional shapes as possible.As inpart a, it may be helpful to vary the angle of rotation of the plane gradually.

1. In the table, describe the shape of the cross section formed when a particular plane passes through the cylinder.

Type your response here:

Description of Plane / Description of Cross Section
plane parallel to the vertical axis
plane parallel to the circular base
plane making an angle with the vertical axis without passing though the base or top surface
plane making an angle with the vertical axis and passing through the base andtop surface
plane making an angle with the vertical axis and passing through either the base or the top surface, but not both

1. Which cross-sectional shapes do you find the most surprising? Explain why.

Type your response here:

1. Select Pyramid from the list of shapes. Set the number of lateral faces to four to explore the cross sections ofrectangular-based pyramids.

1. Vary the plane of intersection, and note all the types of cross sections you observe.In the table, describe the shape of the cross section formed when a particular plane passes through the pyramid.

Type your response here:

Description of Plane / Description of Cross Section
plane perpendicular to the base and passing through the vertical axis
plane parallel to the vertical axis, but not passing through it, and intersecting opposite sides of the base
plane parallel to the vertical axis and intersecting two adjacent sides of the base
plane parallel to the base
plane making an angle with the vertical axis without passing through the base
plane making an angle with the vertical axis and passing through two adjacent sides of the base

1. Which cross-sectional shapes did you find the most surprising? Explain why.

Type your response here:

1. Increase the number of lateral faces of the pyramid, and replicate some of the intersecting planes you used in part ii. What do you notice about the general shape of the cross sections after increasing the number of faces? Explain your observations.

Type your response here:

1. Select Prism from the list of shapes. Set the number of lateral faces to five, and explore the cross sections of a pentagonal prism.Move the intersecting plane, and vary the angle between the intersecting plane and the vertical axis to identify the shapes of different types of cross sections.

1. Note all the types of cross sections you observe.In the table, describe the shape of the cross section formed when a particular plane passes through the prism.

Type your response here:

Description of Plane / Description of Cross Section
plane parallel to the vertical axis
plane parallel to the base
plane making an angle with the vertical axis without passing through the base or the top surface
plane making a sharp angle with the vertical axis and passing through the base and top surface
other planes at various angles with the vertical axis

1. Which cross-sectional shapes did you find the most surprising?

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

1. Rotating Two-Dimensional Objects

You will use an online tool to explore the three-dimensional objects generated when a polygon revolves around a line of rotation. Go to 3d transmographer.

To specify the polygon that you want to rotate, enter the number of vertices of the polygon in the input box, and press Go!Enterthe coordinates of the vertices in order, and press Graph to plotyour polygon. Then build theequation of the line around which you want to rotate your polygon. Once the equation is established, press Revolve, and observe how a solid of revolution (a three-dimensional object)is formed. Changing the polygon or the line of rotationwill changethe solid of revolution.

1. How can you create a cone and a cylinder using this tool? Describe the shapes and the specific axes of rotation that produce a cone and a cylinder. If there is more than one polygon that can generate the same cone or cylinder, state each shape in the table.

Type your response here:

Solid of Revolution / Polygon Rotated / Axis of Rotation
cone
cylinder

1. Experiment with different types of polygons, such as a triangle, rectangle, parallelogram, pentagon, hexagon, and so on, and revolve them around an axis of rotation.Try to create interesting or unusual solids of revolution. Describe atleast two unusual shapes that you can make by revolving a polygon.

Type your response here:

Polygon / Axis of Rotation / Solid of Revolution

1. If the tool allowed it, how would you go about drawing a sphere? Explain.

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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