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Slicing a Right Rectangular Prism with a Plane





 


Videos and solutions to help Grade 7 students learn how to describe rectangular regions that result from slicing a right rectangular prism by a plane perpendicular to one of the faces.

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Common Core For Grade 7

New York State Common Core Math Grade 7, Module 6, Lesson 16


Lesson 16 Student Outcomes


• Students describe rectangular regions that result from slicing a right rectangular prism by a plane perpendicular to one of the faces.

Lesson 16 Summary


• A slice, also known as a plane section, consists of all the points where the plane meets the figure.
• A slice made parallel to a face in a right rectangular prism will be parallel and identical to the face.
• A slice made perpendicular to a face in a right rectangular prism will be a rectangular region with a height equal to the height of the prism.

Lesson 16 Classwork

Provide context to the concept of taking slices of a solid by discussing what comes to mind when we think of “taking a slice” of something.

Example 1
Consider a ball B. Figure 3 shows one possible slice of B.
a. What figure does the slicing plane form? Students may choose their method of representation of the slice (e.g., drawing a 2D sketch, a 3D sketch, or describing the slice in words).
b. Will all slices that pass through be the same size? Explain your reasoning.
c. How will the plane have to meet the ball so that the plane section consists of just one point?

Example 2
The right rectangular prism in Figure 4 has been sliced with a plane parallel to face ABCD. The resulting slice is a rectangular region that is identical to the parallel face.
a. Label the vertices of the rectangular region defined by the slice as WXYZ.
b. To which other face is the slice parallel and identical?

Exercise 1
Discuss the following questions with your group.
1. The right rectangular prism in Figure 5 has been sliced with a plane parallel to face LMON.
a. Label the vertices of the rectangle defined by the slice as RSTU.
b. What are the dimensions of the slice?
c. Based on what you know about right rectangular prisms, which faces must the slice be perpendicular to?



Example 3 The right rectangular prism in Figure 6 has been sliced with a plane perpendicular to BCEH. The resulting slice is a rectangular region with a height equal to the height of the prism.
a. Label the vertices of the rectangle defined by the slice as WXYZ.
b. To which other face is the slice perpendicular?
c. What is the length of ZY?
d. Joey looks at WXYZ and thinks that the slice may be a parallelogram that is not a rectangle. Based on what is known about how the slice is made, can he be right? Justify your reasoning.

Exercises 2–6
In the following exercises, the points at which a slicing plane meets the edges of the right rectangular prism have been marked. Each slice is either parallel or perpendicular to a face of the prism. Use a straightedge to join the points to outline the rectangular region defined by the slice and shade in the rectangular slice.
2. A slice parallel to a face.
3. A slice perpendicular to a face.
4. A slice perpendicular to a face.

In Exercises 5–6, the dimensions of the prisms have been provided. Use the dimensions to sketch the slice from each prism and provide the dimensions of each slice.
5. A slice parallel to a face.
6. A slice perpendicular to a face.


 

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