Topics
8. Mathematical Modeling
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United States of America
Grades 9-12

8.06 Cross sections and revolutions of solids

Lesson
Practice

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.

Cross section

The two-dimensional shape made by slicing through a three-dimensional figure

Apex

The point (vertex) furthest from the base of an object

Exploration

  1. Draw at least two solid figures that would produce circular cross sections.
  2. Draw at least two solid figures that would produce triangular cross sections.
  3. Draw a solid figure that would produce a non-rectangular polygon cross section.
  4. 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.

A cube showing a square cross section.

By taking a slice of a cube that is parallel to one of its sides, we form a cross section that is a square which is congruent to the sides of the cube.

A cube showing a triangle cross section with its vertices passing through the midpoints of three adjacent edges of the cube.

By taking a slice of a cube that passes through the midpoint of three edges, as if we were cutting off a corner of the cube, we form a cross section that is an equilateral triangle in shape.

Examples

Example 1

Find a cross section parallel to the base and identify the shape formed by the cross section.

A pentagonal pyramid.
Worked Solution
Create a strategy

Slice the shape parallel to the base.

A pentagonal pyramid cut by a plane parallel to its base.
Apply the idea

The cross section is the intersection of the shape and the plane. The cross section is a pentagon.

Example 2

Consider the following cross section sliced from a solid figure:

Draw two different figures that the cross section could have come from.

Worked Solution
Create a strategy

We can slice a solid horizontally, vertically, or diagonally in order to get a cross section. We can think of a few triangular-shaped solids that could produce the cross section shown.

Apply the idea

A rectangular pyramid cut vertically from its apex could produce the triangular cross section.

A cone cut vertically could produce the triangular cross section.

Reflect and check

A cube, like the image shown for the equilateral triangle in the lesson could be cut at a steeper angle to produce an isosceles triangle.

Example 3

Consider the cylinder shown below:

A circular cylinder.

Identify three differently shaped cross sections, including at least one that comes from a diagonal slice.

Worked Solution
Create a strategy

Imagine slicing a plane through the cylinder vertically, horizontally, and diagonally at different points to produce differently-shaped cross sections.

Apply the idea

Three shapes that could be formed are a rectangle, an ellipse, and a circle.

Reflect and check

We could also slice a cylinder from one side down to its base, creating a semicircle.

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?

Worked Solution
Create a strategy

If a 3D printer is stacking the layers, we can imagine the cross sections being cut horizontally to the printing surface.

Apply the idea

The printer is printing a cone.

Idea summary

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.

Loading interactive...

  1. 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.

  2. Create a rectangle with one side on the y-axis and drag the slider to 1 \degree. Describe the 3D figure that is formed.

  3. Create another shape and predict what 3D figure will form after rotating it about the y-axis.

  4. 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.

A first and fourth quadrant coordinate plane without numbers with a right triangle plotted on the first quadrant. The base of the triangle is along the positive x axis, and its hypotenuse has a negative slope.
A triangle with one leg along the axis of rotation
A solid of revolution on first and fourth quadrant coordinate plane without numbers. The solid of the revolution resembles a cone in a horizontal position with its center along the x axis.
The resulting solid of revolution

Note that the axis of rotation does not always have to pass through the object to be rotated.

A first and second quadrant coordinate plane without numbers with a rectangle plotted on the second quadrant.
A rectangle away from the axis of rotation
A solid of revolution on a first and second quadrant coordinate plane without numbers. The solid of the revolution resembles a cylinder with a cylindrical hole in its center. The center of the figure is along the positive y axis.
The resulting solid of revolution

Examples

Example 5

Sketch the solid of revolution produced by rotating the following shape about the y-axis:

A four quadrant coordinate plane without numbers with a rectangle plotted on the second and third quadrant. One of the lengths of the rectangle is along the y axis.
Worked Solution
Apply the idea

Reflect the shape over the y-axis.

A four quadrant coordinate plane without numbers with a rectangle plotted on the second and third quadrant, and the same rectangle reflected over the y axis.

Join the end points on the original figure with the corresponding points on the reflection with ovals.

The figure formed by the rectangle after a solid of revolution is a cylinder.

Reflect and check

The solid of revolution produced by rotating the shape about the x-axis would also be a cylinder.

Example 6

Identify the shape that, when rotated about the y-axis, would create the following solid:

Worked Solution
Create a strategy

The shape is rotated about the y-axis, so we find the shape of the cross section that is on the xy-plane. Then, we choose half of the cross section on one side of the y-axis.

Apply the idea

Taking the cross section we get:

A first and second quadrant coordinate plane without numbers. A curved shaped is drawn on the first and second quadrant. The shaped resembles a parabola that opens upward and the vertex is at the origin, and bounded by a horizontal segment on the top.

Taking the left-hand side of the x-axis:

A first and second quadrant coordinate plane without numbers. A curved shaped is drawn on the second quadrant. The shape resembles the left side of a parabola that opens upward and the vertex is at the origin, bounded by a horizontal segment on top, and bounded by a vertical segment along the y axis.
Reflect and check

We would have gotten the same figure if we had used the right half of the cross section.

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.

Worked Solution
Create a strategy

Sketch the two solids formed by rotating the shapes about the each axis.

Right triangle rotated about the y-axis

Right triangle rotated about the x-axis

Apply the idea

By rotating the right triangle about either axis, the solids of revolution will be different.

This is because of the orientation of the right triangle on the coordinate plane. When rotating the figure about the y-axis, a solid cylinder with a hollowed out cone is formed. However, when rotating the figure about the x-axis, a solid cone is formed.

Reflect and check

If the right triangle could be rotated or reflected so that one of its legs aligned with the y-axis, the solids of revolution for each rotation would be identical.

Example 8

A company is designing a new filter for its air purifiers using a computer program. The designer maps out the 2D shape on a coordinate plane, then rotates the shape about the x-axis.

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\text{Feet, }x
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\text{Feet, }y
a

Sketch the filter after its revolution about the x-axis and label its dimensions.

Worked Solution
Create a strategy

The object will be the shape of a cylinder with another cylinder hollowed out from the center.

Apply the idea
b

What is the volume of the air filter?

Worked Solution
Create a strategy

Calculate the volume of the large cylinder as if it was solid. Then subtract the volume of the hollowed out cylinder in the center of the filter, which was created by revolving the 2D shape about the x-axis.

Apply the idea

The cylindrical air filter has a radius of 3 \text{ feet}, so the area of one of its bases is A=\pi \cdot 3^2 \approx 28.27 \text{ ft}^2. To find the volume of the solid, we can multiply the area of the base by the height of the cylinder, so 28.27 \text{ ft}^2 \cdot 3 \text{ ft} = 84.81 \text{ ft}^3.

The cylindrical center of the air filter has a radius of 1 \text{ foot}, so the area of one of its bases is A=\pi \cdot 1^2 \approx 3.14 \text{ ft}^2. To find the volume of the cylindrical center, we can multiply the area of the base by the height of the cylinder, so 3.14 \text{ ft}^2 \cdot 3 \text{ ft} = 9.42 \text{ ft}^3.

The volume of the filter is calculated by finding the difference between the volume of the large, solid cylinder and the volume of the center cylinder. The filter's volume is 84.81 \text{ ft}^3 - 9.42 \text{ ft}^3 = 75.39 \text{ ft}^3.

Reflect and check

Since \pi is an irrational number, we needed to make decisions about how to round it in our calculations. We should take the units we're working with and prractical considerations into account. Since the measurements are in feet and cubic feet, rounding to the hundredths place is reasonable, and likely within the capabilities of a computer design program and necessary level of accurate measurements for an air filter. We could also opt not to round, and instead use calculator software at each step, which uses more exact calculations of \pi, which will result in a more precise solution.

Once we make the decision to round to the hundredths place, that becomes the limit of our level of precision, and our final solution should not be given with more than two decimal placements.

Idea summary

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.

Outcomes

G.GMD.B.4

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

G.MG.A.1

Use geometric shapes, their measures, and their properties to describe objects.

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