Lesson 11Slicing Solids

Learning Goal

Let’s see what shapes you get when you slice a three-dimensional object.

Learning Targets

  • I can explain that when a three dimensional figure is sliced it creates a face that is two dimensional.

  • I can picture different cross sections of prisms and pyramids.

Lesson Terms

  • base (of a prism or pyramid)
  • cross section
  • prism
  • pyramid

Warm Up: Prisms, Pyramids, and Polyhedra

Problem 1

Describe each shape as precisely as you can. Click on the applet and drag the mouse to show the object turning in 3D.

  1. open applet in presentation mode
  2. open applet in presentation mode
  3. open applet in presentation mode
Print Version

Describe each shape as precisely as you can.

  1. a green triangular prism with a base that is a right triangle.
  2. a blue rectangular pyramid with a vertex that is not centered over the base
  3. a red prism with a base that is a pentagon

Activity 1: What’s the Cross Section?

Problem 1

Here are a rectangular prism and a pyramid with the same base and same height. Drag the large red point up and down to move the plane through the solids.

open applet in presentation mode

  1. If we slice each solid parallel to its base halfway up, what shape cross sections would we get? What is the same about the cross sections? What is different?

  2. If we slice each solid parallel to its base near the top, what shape cross sections would we get? What is the same about the cross sections? What is different?

Print Version

Here is a rectangular prism and a pyramid with the same base and same height.

A rectangular prism and pyramid with a base of 2 cm and and height of 4 cm.

  1. Think about slicing each solid parallel to its base, halfway up. What shape would each cross section be? What is the same about the two cross sections? What is different?

  2. Think about slicing each solid parallel to its base, near the top. What shape would each cross section be? What is the same about the two cross sections? What is different?

Are you ready for more?

Problem 1

Describe the cross sections that would result from slicing each solid perpendicular to its base.

Activity 2: Card Sort: Cross Sections

Problem 1

Your teacher will give you a set of cards. Sort the images into groups that make sense to you. Be prepared to explain your reasoning.

Activity 3: Drawing Cross Sections

Problem 1

Use the applet to draw each cross section and describe it in words.

Here is an applet with a rectangular prism, 4 units by 2 units by 3 units.

open applet in presentation mode

  1. A plane cuts the prism parallel to the bottom and top faces.

  2. The plane moves up and cuts the prism at a different height.

  3. A vertical plane cuts the prism diagonally.

Print Version

Draw and describe each cross section.

Here is a picture of a rectangular prism, 4 units by 2 units by 3 units.

A rectangular prism 4 by 2 by 3 units

  1. A plane cuts the prism parallel to the bottom and top faces.

    A plane cutting the prism parallel to the bottom and top faces.

  2. The plane moves up and cuts the prism at a different height.

    The plane moves up and cuts the prism at a different height.

  3. A vertical plane cuts the prism diagonally.

    A vertical plane cuts the prism diagonally.

Problem 2

Use the applet to draw each cross section and describe it in words.

A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units.

open applet in presentation mode

  1. A plane cuts the pyramid parallel to the base.

  2. A vertical plane cuts the prism.

Print Version

Draw and describe each cross section.
A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units.

  1. A plane cuts the pyramid parallel to to the base

    A square pyramid with a base 4 by 4 units and a height of 4 units. A plane cuts the pyramid parallel to the base.

  2. A vertical plane cuts the prism.

    A vertical plane cuts the prism

Problem 3

Draw and describe each cross section.

A cube has an edge of length 4.

open applet in presentation mode

  1. A plane cuts off the corner of the cube.

  2. The plane moves farther from the corner and makes a cut through the middle of the cube.

Print Version

Draw and describe each cross section.
A cube has an edge of length 4.

  1. A plane cuts off the corner of the cube.

    A cube with an edge length of 4. A place cuts off the corner of the cube

  2. The plane moves farther from the corner and makes a cut through the middle of the cube.

    The plane moves farther from the corner and makes a cut through the middle of the cube

Lesson Summary

When we slice a three-dimensional object, we expose new faces that are two dimensional. The two-dimensional face is a cross section. Many different cross sections are possible when slicing the same three-dimensional object.

Here are two peppers. One is sliced horizontally, and the other is sliced vertically, producing different cross sections.

Three images are indicated. The first image is of two whole peppers. In the second image, the first pepper is sliced in half horizontally and the second pepper is sliced in half vertically. The third image is of each cross section created by the slices and the painted imprint of each cross section.

The imprints of the slices represent the two-dimensional faces created by each slice.

It takes practice imagining what the cross section of a three-dimensional object will be for different slices. It helps to experiment and see for yourself what happens!