Area of an Ellipse

 

     Suppose that we have an ellipse with major axis length 2a and minor axis length 2b.

 

     Without changing the answer to the problem, suppose that the major axis is on the x-axis.

 

     This is all illustrated in the figure to the left. The question remains: how do we find the area of the ellipse?

 

 

 

 

     First, notice that if we can find the area in the first quadrant (see the shaded area in the figure below), we can multiple it by four to find the total area. But how can we find the area in the first quadrant? To do that, we need to recall the equation of the ellipse.

 

     Thankfully, with a background in pre-calculus this is not too difficult. Recall that the equation of an ellipse like the one picture above is given by:

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     Now, we need to solve for y. Doing so, we get:

 

      

 

     Since we are looking for the area in the first quadrant, we only need to use the positive square root. Now that we have the function, we can set up an integral. Notice that we are integrating the function  from 0 to a. Set as an integral, we have:

      

 

     To evaluate this integral, notice that we can pull out the b/a, and that we will have to use a trigonometric substitution. Let x = asinθ. That means that dx = acosθdθ. Also, our limits of integration change with this substitution. We start at 0 = asinθ which implies that θ = 0 and end at a = asinθ which implies that θ = π / 2. Plugging that in, we have:

 

 

      

 

     And so, the area of the first quadrant is . Recall earlier that we said that the area of the whole ellipse would be 4 times the area of the first quadrant, and thus the area of the entire ellipse is abπ.

 

 

     Note: The area of a circle is a special case of this general formula. When dealing with a circle, both a and b are equal to r. If we plug those into the above formula, we see that the area of the circle is , as it should be.