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Lesson Plans and Worksheets for Geometry

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More Lessons for Geometry

Common Core 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 πΈ and πΈβ² be two parallel planes. Let π΅ be a rectangular region1
in the plane πΈ. At each
point π of π΅, consider ππβ perpendicular to πΈ, joining π to a point πβ² of the plane πΈβ². The union of all these segments is
called a right rectangular prism.

**GENERAL CYLINDER**: (See Figure 1.) Let πΈ and πΈβ² be two parallel planes, let π΅ be a region2
in the plane πΈ, and let πΏ be a line
that intersects πΈ and πΈβ² but not π΅. At each point π of π΅, consider ππβ² parallel to πΏ, joining π to a point πβ² of the plane πΈβ². 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 πΈ and πΈβ² be two parallel planes. Let π΅ be a rectangular region in the plane πΈ. At each
point π of π΅, consider ππβ² perpendicular to πΈ, joining π to a point πβ² of the plane πΈβ². 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 ππ
is a vertex of π΅. Let ππβ² be the corresponding point in π΅β² such that ππππβ²is parallel to the line πΏ defining the prism. ππππβ² is called a lateral
edge of the prism. If ππππ+1 is a base edge of the base π΅ (a side of π΅), and πΉ is the union of all segments ππβ² parallel
to πΏ for which π is in ππππ+1 and πβ² is in π΅β², then πΉ 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 πΈ and πΈβ² be two parallel planes, let π΅ be a region in the plane πΈ, and let πΏ be a line that
intersects πΈ and πΈβ² but not π΅. At each point π of π΅, consider ππβ² parallel to πΏ, joining π to a point πβ² of the plane πΈβ². The union of all these segments is called a general cylinder with base π΅.

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