General Prisms and Cylinders and Their Cross-Sections

New York State Common Core Math Geometry, Module 3, Lesson 6

Worksheets for Geometry

Student Outcomes

• Students understand the definitions of a general prism and a cylinder and the distinction between a cross-section and a slice.

General Prisms and Cylinders and Their Cross-Sections

Classwork

Opening Exercise

Sketch a right rectangular prism.

RIGHT RECTANGULAR PRISM: Let 𝐸 and 𝐸′ be two parallel planes. Let 𝐵 be a rectangular region1 in the plane 𝐸. At each point 𝑃 of 𝐵, consider 𝑃𝑃’ perpendicular to 𝐸, joining 𝑃 to a point 𝑃′ of the plane 𝐸′. The union of all these segments is called a right rectangular prism.

GENERAL CYLINDER: (See Figure 1.) Let 𝐸 and 𝐸′ be two parallel planes, let 𝐵 be a region2 in the plane 𝐸, and let 𝐿 be a line that intersects 𝐸 and 𝐸′ but not 𝐵. At each point 𝑃 of 𝐵, consider 𝑃𝑃′ parallel to 𝐿, joining 𝑃 to a point 𝑃′ of the plane 𝐸′. The union of all these segments is called a general cylinder with base 𝐵.

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Discussion

Example of a cross-section of a prism, where the intersection of a plane with the solid is parallel to the base.

A general intersection of a plane with a prism, which is sometimes referred to as a slice.

Exercise

Sketch the cross-section for the following figures:

Lesson Summary

RIGHT RECTANGULAR PRISM: Let 𝐸 and 𝐸′ be two parallel planes. Let 𝐵 be a rectangular region in the plane 𝐸. At each point 𝑃 of 𝐵, consider 𝑃𝑃′ perpendicular to 𝐸, joining 𝑃 to a point 𝑃′ of the plane 𝐸′. The union of all these segments is called a right rectangular prism.

LATERAL EDGE AND FACE OF A PRISM: Suppose the base 𝐵 of a prism is a polygonal region, and 𝑃𝑖 is a vertex of 𝐵. Let 𝑃𝑖′ be the corresponding point in 𝐵′ such that 𝑃𝑖𝑃𝑖′is parallel to the line 𝐿 defining the prism. 𝑃𝑖𝑃𝑖′ is called a lateral edge of the prism. If 𝑃𝑖𝑃𝑖+1 is a base edge of the base 𝐵 (a side of 𝐵), and 𝐹 is the union of all segments 𝑃𝑃′ parallel to 𝐿 for which 𝑃 is in 𝑃𝑖𝑃𝑖+1 and 𝑃′ is in 𝐵′, then 𝐹 is a lateral face of the prism. It can be shown that a lateral face of a prism is always a region enclosed by a parallelogram.

GENERAL CYLINDER: Let 𝐸 and 𝐸′ be two parallel planes, let 𝐵 be a region in the plane 𝐸, and let 𝐿 be a line that intersects 𝐸 and 𝐸′ but not 𝐵. At each point 𝑃 of 𝐵, consider 𝑃𝑃′ parallel to 𝐿, joining 𝑃 to a point 𝑃′ of the plane 𝐸′. The union of all these segments is called a general cylinder with base 𝐵.

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