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Extending to Three Dimensions

Rotation of Objects in Two Dimensions

This lesson aims to show that three-dimensional objects can be generated by rotations of two-dimensional objects. Additionally, the shapes of two-dimensional cross-sections of three-dimensional objects will be identified and analyzed.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Cross-Sections of a Cube

In the diagram, a cube is given. Try to identify how many different cross-sections can be formed. What geometric shapes do these cross-sections have?

A cube
Can a triangle be formed? Can a pentagon be formed? What other polygons can be formed?

Explore

Rotating a Circle

Consider the given circle. Rotate it about its diameter and try to identify the 3D object formed.
Rotating a circle about its diameter
Think of whether it is the only possible object that can be formed when rotating a circle about its diameter. Can other three-dimensional objects be formed?

Extra

How to Use the Applet

These are the possible interactions with the presented applet.

  • To rotate the circle, click on the button Rotate or click on the circle and move it around.
  • To see the formed three-dimensional figure from different perspectives, click on its interior region and drag it.
  • To return to initial image, click on the button Reset.

Discussion

Cross-Sections of a Sphere

Previously, it was observed that rotating a circle around its diameter creates a sphere. Now different types of cross-sections of a sphere will be identified.
A sphere intersected by a plane
As can be noted, every cross-section of a sphere is a circle. Although some cross-sections appear to be ellipses from different points of view, keep in mind that objects may look different from their actual shape depending on the perspective.

Extra

How to Use the Applet

These are possible interactions with the presented applet.

  • To raise or lower the intersecting plane, move the slider on the right either up or down.
  • To rotate the plane about the center of the sphere, click and drag the Rotation point.
  • To see the sphere intersected by the plane from different perspectives, click on the sphere's interior region and drag it.

Explore

Rotating a Rectangle

Consider a rectangle. Choose a side to be the axis of rotation by clicking on it and then rotate the rectangle.
A rectangle rotated about one its side forming a right cylinder
What type of 3D figure is formed? Would the formation of such a figure occur only in this one case, or can it be said to be a general rule?

Extra

How to Use the Applet

These are the possible interactions with the presented applet.

  • To choose the side to be the axis of rotation, click on the longer or shorter side.
  • To rotate the rectangle, click on the Rotate button.
  • To look at the formed three-dimensional figure from different perspectives, click on its interior region and drag it.
  • To return to initial image, click on the Reset button.

Discussion

Cross-Sections of a Right Cylinder

From the previous applet, it could be concluded that rotating a rectangle about one of its sides forms a right cylinder. What are the cross-sections of a right cylinder? There are several types depending on the position of an intersecting plane.

Case Position of the Plane Cross-Section
Perpendicular to the base Rectangle
Parallel to the base Circle
Diagonal to the base Ellipse

The following applet illustrates each type of the mentioned cross-sections.

Three types of cross-sections of a right cylinder

Example

Pairing a Plane with a Cylinder's Cross-Section

In the diagram is a right cylinder that is intersected by three planes. Pair each plane with the corresponding shape of the cross-section formed.
Three types of cross-sections of cylinder

Hint

Analyze the shapes of the cross-sections closely. What conclusions can be drawn about the sides and angles of the cross-sections?

Solution

To pair each plane with the corresponding type of cross-section, each given case should be investigated.

Plane A

The first intersecting plane is parallel to the cylinder's bases. Therefore, the cross-section it creates has the same shape as the bases, which are circles.

Parallel to the base plane intersecting cylinder

Hence, the cross-section is also a circle.

Plane B

The intersecting plane is perpendicular to both bases of the cylinder. Since the cylinder is right, the cross-section has four right angles. Additionally, the longer sides of the cross-section are parallel to the height of the cylinder and have the same lengths.

Perpendicular to the base plane intersecting cylinder

Furthermore, the shorter sides are parallel chords of the cylinder's bases, which also have equal lengths. Therefore, the cross-section is a rectangle.

Plane C

The last plane intersects only the curved surface of the cylinder, so it does not have straight sides. The cross-section has a circular shape, but it is not a circle.

Diagonal to the base plane intersecting cylinder

Recall the possible cross-sections of a right cylinder depending on the position of the intersecting plane.

The Plane's Position Cross-Section
Perpendicular to the base Rectangle
Parallel to the base Circle
Diagonal to the base Ellipse

Since the intersecting plane is diagonal to the bases of the cylinder, it can be concluded that the cross-section is an ellipse.

Pop Quiz

Identifying the Types of a Cylinder's Cross-Sections

The given two-dimensional shape is a cross-section of a right cylinder. Identify whether the cross-section is parallel, perpendicular, or diagonal to the base of the cylinder.

Different types of cross-sections of a right cylinder

Explore

Rotating a Right Triangle

Here a right triangle is given. Rotate it about its height and identify the type of a three-dimensional figure formed.
A right triangle rotated about its height forming a cone
Think about whether this is a one-time case or a general rule.

Extra

How to Use the Applet

These are the possible interactions with the presented applet.

  • To choose the axis of rotation of a triangle, click on either of its two legs (each of them can be treated as the height of the triangle).
  • To rotate the triangle, click on the button Rotate.
  • To see the formed three-dimensional figure from different perspectives, click on its interior region and drag it.
  • To return to the initial image, click on the button Reset.

Discussion

Cross-Sections of a Cone

From the previous applet, it can be observed that when rotating a right triangle about its height, a right cone is formed. What are the cross-sections of a right cone? Here are some possible types depending on the position of an intersecting plane.

The Plane's Position Cross-Section
Perpendicular to the base Triangle
Parallel to the base Circle
Diagonal to the base Ellipse
The following applet illustrates each type of the mentioned cross-section.
Cone cross-sections
Please note that in the case of an ellipse cross-section, the intersecting plane can be positioned parallel to the base, which turns the ellipse into a circle.

Example

Pairing a Plane with a Cone's Cross-Section

In the diagram is a right cone that is intersected by three planes. Pair each plane with the corresponding shape of the cross-section formed.
Three types of cross-sections of a right cone

Hint

Analyze the shapes of the cross-sections closely. What conclusions can be drawn about the sides and angles of the cross-sections?

Solution

To pair each plane with the corresponding type of cross-section, each given case should be analyzed.

Plane A

The first intersecting plane is parallel to the cone's base. Consequently, the cross-section it creates has the same shape as the base, which is a circle.

Circle cross-section

Therefore, the cross-section is also a circle.

Plane B

The second plane goes through the curved surface of the cone, so it does not have any straight sides. The cross-section has a circular shape, but it is not a circle.

Ellipse cross-section

To identify the cross-section, the possible cross-sections of a right cone can be reviewed.

The Plane's Position Cross-Section
Perpendicular to the base Triangle
Parallel to the base Circle
Diagonal to the base Ellipse

It can be noticed that the intersecting plane is diagonal to the bases of the cone. Therefore, the cross-section can be concluded to be an ellipse.

Plane C

The last intersecting plane is perpendicular to the base of the cone and goes through the vertex of the cone. The cross-section has three vertices, which are connected by sides.

Triangle cross-section

Therefore, the cross-section is a triangle.

Pop Quiz

Identifying Cross-Sections of a Cone

The given 2D shape is a cross-section of a right cone. Identify whether the cross-section is parallel, perpendicular, or diagonal to the base of the cone.

Different types of cross-sections of a right cone

Closure

Other Possible Cross-Sections of a Cone

This less analyzed what types of 3D figures are formed by rotating different 2D objects, such as a circle, a rectangle, and a right triangle.
Three three-dimensional figures
Also, different types of cross-sections of a sphere, a right cylinder, and a right cone were investigated.
3D Object Intersecting Plane's Position Cross-Section
Sphere Any position Circle
Right Cylinder Perpendicular to the base Rectangle
Parallel to the base Circle
Diagonal to the base Ellipse
Right Cone Perpendicular to the base Triangle
Parallel to the base Circle
Diagonal to the base Ellipse

Are these all the possible cross-sections? What if a plane goes through a cone in such way that it is diagonal to the base and intersects the base? Or what if it is perpendicular to the base but does not go through the vertex of the cone? These are two more possible cross-sections of a cone.

Hyperbola and parabola cross-sections of a cone


As can be noted, the cross-sections have a shape of a hyperbola and a parabola. Wow, this realm of 3D figures, planes, and cross-sections are fascinating! Try to investigate whether these cross-sections also apply to a right cylinder.
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