Cross Section: Definition & Example

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will define cross sections. We will look at famous types of cross sections called conic sections, how cross sections show up in the world around us, and how the study of these is useful in various real world scenarios.

Cross Sections

Suppose your day is just getting started and you decide to have some toast for breakfast. To make your toast, you take a loaf of bread and slice off a piece. Take a look at the shape of the slice that you created. In mathematics, we call this a cross section. You just made a cross section of your bread loaf but cutting off a slice!


Cross Section
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A cross section is the shape that you create when you cut through an object. In the case of the loaf of bread, the cross-section is in the shape of a piece of bread.

Conic Sections

The first study of cross sections can be dated to around 360-350 BC in ancient Greece. Around this time a Greek mathematician named Menaechmus discovered some specific types of cross sections called conic sections. Conic sections are some of the most well-known cross sections and certainly deserve mention here, as they will really help us to understand and analyze cross sections in general.

To explain conic sections, let's first define a right cone. A right cone is a three-dimensional object that has a circular base at one end and a point, called the apex, at the other. The apex lies directly above the center of the circular base.


Right Cone
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Now, let's define conic sections. Conic sections are the cross sections we create by slicing a right cone in various ways. There are four types of conic sections. There is a circle, an ellipse, a parabola, and a hyperbola.


Conic Sections
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Each of these shapes is a cross section created by slicing a right cone at different angles.


Conic Section Slices
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These conic sections nicely illustrate cross sections and how cross sections are created. This is all fine and dandy, but how does this help us in real life? Let's consider this question.

Cross Sections in Real Life

As it turns out, cross sections not only lend themselves to the study of geometry and shapes, they also show up quite often in the world around us. The fact that they lend themselves to the study of shapes and that these shapes are all around us tells us that cross sections can help us analyze the various characteristics of the inside of an object as well as how it is put together.

For example, consider the Tycho Brahe Planetarium in Copenhagen. This building is in the shape of a vertical tube that has been sliced diagonally, creating an ellipse as the roof of the building.


Cross Section in Real Life
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Cross sections are used in architecture and engineering quite often. By understanding cross sections and how to cut a certain object to create the desired shape, we are able to construct amazing buildings and other inventions.

Another common example of cross sections are the cross sections created when sawing wood. Based on the log, we can create the desired shapes (cross sections) by slicing it in different ways. These cross sections can be examined to determine the age or health of the tree.


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