6. The Rotation of Two Dimensional Geometric Objects
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Chapter 3
6.
The Rotation of Two Dimensional Geometric Objects
This lesson explores the concept of rotating two-dimensional geometric shapes to create three-dimensional objects. It delves into the rotation of circles, rectangles, and right triangles, explaining how these actions result in spheres, cylinders, and cones, respectively. The lesson also discusses the types of cross-sections that can be formed when a plane intersects these three-dimensional figures. For instance, a sphere always has circular cross-sections, while a cylinder can have circular, rectangular, or elliptical cross-sections depending on the angle of intersection. Understanding these principles is crucial for fields like engineering, architecture, and computer graphics, where the manipulation of shapes in space is often required.
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Lesson Settings & Tools
| | 13 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
The Rotation of Two Dimensional Geometric Objects
Slide of 13
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.
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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?
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Rotating a Circle
Consider the given circle. Rotate it about its diameter and try to identify the 3D object formed.
Extra
How to Use the AppletThese 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.
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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.
Extra
How to Use the AppletThese are possible interactions with the presented applet.
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Rotating a Rectangle
Consider a rectangle. Choose a side to be the axis of rotation by clicking on it and then rotate the rectangle.
Extra
How to Use the AppletThese 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.
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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 |
|---|---|---|
| 1 | Perpendicular to the base | Rectangle |
| 2 | Parallel to the base | Circle |
| 3 | Diagonal to the base | Ellipse |
The following applet illustrates each type of the mentioned cross-sections.
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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.
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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.
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.
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.
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.
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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.
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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.
Extra
How to Use the AppletThese 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.
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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.
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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.
{"type":"pair","form":{"alts":[[{"id":0,"text":"A"},{"id":1,"text":"B"},{"id":2,"text":"C"}],[{"id":0,"text":"Triangle"},{"id":1,"text":"Circle"},{"id":2,"text":"Ellipse"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[1,2,0]]}
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.
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.
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.
Therefore, the cross-section is a triangle.
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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.
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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.
| 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.
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The Rotation of Two Dimensional Geometric Objects
Exercise 3.1
What is the surface area of the solid created when the triangle is rotated around the y-axis? Give the answer in terms of π.
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When we revolve the triangle about the y-axis, we get a solid of revolution that is a cone. However, the given triangle will produce a solid with a cone-shaped hole in the middle. In the following diagram, we see the resulting solid after completing the rotation about the y-axis.
The surface area of this composite solid is the surface area of the big cone, minus the circular area in the center, plus the lateral surface area of the hole. To calculate the surface area of a cone, we need its radius and slant height.
Finding the Slant Heights
The big cone has a radius and height of 4 and 8 centimeters. The small cone has a radius and height of 1 and 8 centimeters. With this information, we can calculate the slant heights of each cone using the Pythagorean Theorem. Big Cone: (l_B)^2=4^2+8^2 ⇒ l_B = 4sqrt(5) [0.1em] Small Cone: (l_S)^2=1^2+8^2 ⇒ l_S =sqrt(65) Now we can calculate the areas we need.
Surface Area of the Big Cone
Let's calculate the surface area of the big cone SA_B as if it did not have a hole.
Surface Areas of the Small Cone
As previously explained, to obtain the surface area of the composite solid, we have to add the lateral surface area of the hole and subtract the base area. Let's determine these parts separately. Base area:& π (1)^2 = π Lateral area:& π (1)(sqrt(65))= sqrt(65)π
Surface Area of the Composite Solid
Now we can find the surface area of the composite solid. SA_(CS)= 16π+16sqrt(5)π- π+ sqrt(65)π ⇓ SA_(CS)=15π+16sqrt(5)π+sqrt(65)π The surface area of the composite solid is 15π+16sqrt(5)π+sqrt(65)π square units.
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The Rotation of Two Dimensional Geometric Objects
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