9.02 Cross sections and revolutions of solids
Introduction
We were introduced to cross sections of three-dimensional solids in 7th grade. This lesson will extend that concept to draw and identify the cross sections and three-dimensional solids formed by various cross sections. We will explore how 2D shapes can be rotated on the coordinate plane to form 3D solids.
Cross sections of solids
A solid is a term used when talking about a three-dimensional object.
By slicing through a solid, we produce a two-dimensional shape called a cross section. The figure created from a cross section depends on the orientation or angle of the intersecting plane.
Exploration
- Draw at least two solid figures that would produce circular cross sections.
- Draw at least two solid figures that would produce triangular cross sections.
- Draw a solid figure that would produce a non-rectangular polygon cross section.
- What additional information would be useful in deciding what three-dimensional solid to draw from the cross section shape?
A solid may form many different shapes by taking different cross sections. In particular, knowing a cross section of a solid isn't enough information to uniquely determine the original solid. Many different solids can produce identical cross sections.
Cross sections are formed when taking a slice of a figure horizontally, vertically, or diagonally. The shapes that form from cross sections may be more or less obvious, like when we take a vertical slice of a cube versus a diagonal slice.
Examples
Example 1
Find a cross section parallel to the base and identify the shape formed by the cross section.
Example 2
Consider the following cross section sliced from a solid figure:
Draw two different figures that the cross section could have come from.
Example 3
Consider the cylinder shown below:
Identify three differently shaped cross sections, including at least one that comes from a diagonal slice.
Example 4
A 3D printer uses computer assistance to stack layers of material that make a three-dimensional shape. The printer creates an object out of several layers to create a physical model of a computer image. Shown below are the layers of a model that a 3D printer has created. What solid figure is created by the printer?
A single three-dimensional solid may have cross sections of different shapes, depending on how the solid is sliced (vertical, horizontal or diagonal).
Solids of revolution
Exploration
Explore the applet by dragging the points on the shape and the slider.
Create a right triangle with one leg on the y-axis and drag the slider to 1 \degree. Describe the 3D figure that is formed.
Create a rectangle with one side on the y-axis and drag the slider to 1 \degree. Describe the 3D figure that is formed.
Create another shape and predict what 3D figure will form after rotating it about the y-axis.
Describe what is happening with each shape created.
When a two-dimensional object is rotated about an axis it forms a three dimensional object called a solid of revolution.
A revolution is a full turn around an axis, or rotation around a point. The angular measure of one revolution is 360 \degree.
Note that the axis of rotation does not always have to pass through the object to be rotated.
Examples
Example 5
Sketch the solid of revolution produced by rotating the following shape about the y-axis:
Example 6
Identify the shape that, when rotated about the y-axis, would create the following solid:
Example 7
Consider the figure on the coordinate plane shown below:
If the right triangle is rotated about the y-axis or the shape is rotated about the x-axis, will the solids of revolution formed be identical? Explain how you know.
A solid of revolution is formed by rotating a shape about the x- or y-axis, resulting in solids that may be the same or different depending on the orientation of a figure on the coordinate plane.