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.
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?
Explore
Rotating a Circle
Consider the given circle. Rotate it about its diameter and try to identify the 3D object formed.
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 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.
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.
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 AppletThese are possible interactions with the presented applet.
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. 
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 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.
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.
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.
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.
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.
Think about whether this is a one-time case or a general rule.
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.
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 |
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.
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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 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.
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.
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.
Also, different types of cross-sections of a sphere, a right cylinder, and a right cone were investigated.
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.
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.
| 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 |
The Rotation of Two Dimensional Geometric Objects
Exercise 2.1
The following figure shows a cube with a side length of 5 inches and a cross-section through four of its vertices.
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The cross-section of the cube consists of two sides and two diagonals of the cube. Let's mark them on the diagram.
Knowing that the cube has sides of 5 inches, we can find the length of the diagonal by using the Pythagorean Theorem.
Now that we know the length of the diagonal, we can find the perimeter of the cross-section by adding the lengths of the sides.
The perimeter is 10+10sqrt(2) inches.
To find the area of the rectangular cross-section, we will multiply the length and width of the cross-section.
The area is 25sqrt(2) square inches.
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Consider the following cube.
ACircleBPentagonCRhombusDIsosceles TriangleENone
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Let's investigate the potential cross-sections of the cube one at the time.
A. Circle
Let's recall the definition of a circle.
A circle is the set of all points that are equidistant from a given point.
From this definition, we can conclude that a circle has a rounded shape. A cube, though, is a polyhedron. Regardless of how the plane intersects the cube, the resulting shape will always be a polygon. Therefore, we know that it is not possible for a cross-section of a cube to be a circle.
B. Pentagon
As we previously discussed, no matter how the plane intersects the cube, the result will always be a polygon. If the plane passes through four edges and a vertex of the cube, we have a cross-section in the shape of a pentagon.
C. Rhombus
Notice that a rhombus can be a square if all the angles are right. Therefore, we can create a rhombus with a cross-section that runs parallel any face of the cube.
D. Isosceles Triangle
An isosceles triangle is a triangle with a pair of congruent sides. If we let the plane pass through the three edges of a common vertex where two points of intersection are at the same distance from the vertex, we will get an isosceles triangle.
Conclusion
In conclusion, the cross-section of a cube can create a pentagon, a rhombus, and an isosceles triangle. & A Circle && * & B Pentagon&& ✓ & C Rhombus&& ✓ & D Isosceles Triangle && ✓ & E None && *
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Consider the following figure.
By rotating the shape around the axis, we get a composite solid. Determine the following measures of this solid in terms of π.
Volume
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Surface Area
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By revolving the figure about the axis, we get a composite solid that consists of two cones attached at their base.
The solid consists of two identical cones with a height of 4 inches and a radius of 3 inches. To determine the volume of a cone, we calculate one-third of the base area times its height. V=1/3π r^2 h Since we have two cones, we must multiply this formula by 2 to get the volume of the composite solid V_(CS).
The volume of the composite solid is 24π cubic inches.
The surface area of the composite solid is twice the surface area of a cone minus their base area. In other words, we only want to calculate the sum of the cone's lateral surfaces.
SA_(CS)=2(π r l)
To calculate the lateral surface, we need the slant height of the cone. As we can see, the slant height is the hypotenuse of a right triangle with legs of 3 and 4 inches. The measurements of such a triangle form a Pythagorean Triple, so the triangle has a hypotenuse of 5 inches.
Let's substitute the radius and slant height into the formula and evaluate the right-hand side.
The surface area of the composite solid is 30π square inches.
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The Rotation of Two Dimensional Geometric Objects
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