Unlocking the Mysteries of Rotation in the Coordinate Plane and Beyond

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?

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

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.

Three types of cross-sections of a right cylinder

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.

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.

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