# General Prisms and Cylinders and Their Cross-Sections

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

Worksheets for Geometry, Module 3, Lesson 6

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 πΈ and πΈβ² be two parallel planes. Let π΅ be a rectangular region1 in the plane πΈ. At each point π of π΅, consider ππβ perpendicular to πΈ, joining π to a point πβ² of the plane πΈβ². The union of all these segments is called a right rectangular prism.

GENERAL CYLINDER: (See Figure 1.) Let πΈ and πΈβ² be two parallel planes, let π΅ be a region2 in the plane πΈ, and let πΏ be a line that intersects πΈ and πΈβ² but not π΅. At each point π of π΅, consider ππβ² parallel to πΏ, joining π to a point πβ² of the plane πΈβ². The union of all these segments is called a general cylinder with base π΅.

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 πΈ and πΈβ² be two parallel planes. Let π΅ be a rectangular region in the plane πΈ. At each point π of π΅, consider ππβ² perpendicular to πΈ, joining π to a point πβ² of the plane πΈβ². The union of all these segments is called a right rectangular prism.

LATERAL EDGE AND FACE OF A PRISM: Suppose the base π΅ of a prism is a polygonal region, and ππ is a vertex of π΅. Let ππβ² be the corresponding point in π΅β² such that ππππβ²is parallel to the line πΏ defining the prism. ππππβ² is called a lateral edge of the prism. If ππππ+1 is a base edge of the base π΅ (a side of π΅), and πΉ is the union of all segments ππβ² parallel to πΏ for which π is in ππππ+1 and πβ² is in π΅β², then πΉ 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 πΈ and πΈβ² be two parallel planes, let π΅ be a region in the plane πΈ, and let πΏ be a line that intersects πΈ and πΈβ² but not π΅. At each point π of π΅, consider ππβ² parallel to πΏ, joining π to a point πβ² of the plane πΈβ². The union of all these segments is called a general cylinder with base π΅.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.