# Semi-Major Axis of an Ellipse

Instructor: Laura Pennington

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

This lesson will define an ellipse, its semi-major axis, and a few other characteristics of an ellipse. We will also see how to find the length of the semi-major axis of an ellipse and solve an application involving this process.

## The Definition of an Ellipse

Are you familiar with the shape of an ellipse? Have you ever eaten a burrito? Wait, what? What do these two things have to do with each other? It just so happens that they do have something to do with each other! You see, the end of a burrito takes on the shape of an ellipse!

Notice that the end of the burrito looks like a circle that's been squashed down a bit? Well, that is exactly how an ellipse looks!

Of course, a squashed down circle is not the technical definition of an ellipse! To be more formal, we define an ellipse as the set of all points with distance from two fixed points, F and G, adding up to the same number. We call the two fixed points, F and G, the foci of the ellipse. Even more formally, we say that an ellipse is the set of all points, A, such that if F and G are the foci of the ellipse, then AF + AG is constant.

That's a pretty technical definition, but good to know even if we do prefer to just think of an ellipse as a circle that has been squished down a little.

## Semi-Major Axis of an Ellipse

Now that we know what an ellipse is, let's talk about some of its parts. In particular, we want to talk about the semi-major axis of an ellipse. However, to introduce the semi-major axis of an ellipse, we must first recognize the major axis of an ellipse!

If we were to place an ellipse on an xy-axis, with the origin at the center of the ellipse, one of the axes inside the ellipse would be a bit longer than the other, depending on if the circle was squished vertically or horizontally to create the ellipse.

Looking at an ellipse in this way, we call the two axes inside the ellipse the major axis and the minor axis of the ellipse. The major axis is the longer axis, and the minor axis is the shorter axis.

We can now define the semi-major axis of an ellipse as half of the major axis of an ellipse!

So far, so good! Nothing too complicated! Let's keep that going!

## Finding the Semi-Major Axis of an Ellipse

Now that we know what the semi-major axis of an ellipse is, let's talk about finding its length! We know that it is half of the major axis, so its length would be half of the length of the major axis, and as it turns out, the length of the major axis can be found by simply adding up the distance from one foci, F, to any point, A, on the ellipse and the distance from the other foci, G, to the same point A.

We see that AF + AG is equal to the length of the major axis. Therefore, we have the following;

Length of the major axis = AF + AG, where F and G are the foci of the ellipse, and A is any point on the ellipse.

Well, if that is the length of the major axis, then to find the length of the semi-major axis, we simply split it in half, or divide it by two!

Length of the semi-major axis = (AF + AG) / 2, where F and G are the foci of the ellipse, and A is any point on the ellipse.

That's pretty easy! Let's try putting this formula into action!

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