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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?

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 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

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.