In this article, you will learn everything you need to know about the special shape ellipse and also how to calculate its area.

Shall we begin?

In this article, you will learn everything you need to know about the special shape ellipse and also how to calculate its area.

Shall we begin?

This is our ellipse:

*Illustration in Word file*

On the ellipse, we will draw the axes $X$ and $Y$ in order to better understand the material.

The canonical equation of the ellipse (with its center at $0,0$) is:

$\frac{x^2}{a^2} +\frac{y^2}{b^2} =1$

The intersection points of the ellipse with the $X$ axis are:

$(a,0)$ and – $( -a,0)$

The points of intersection of the ellipse with the $Y$ axis are:

$(0 ,b)$ and $(0 ,-b)$

The foci of the ellipse are:

$(c, 0)$ and $(-c, 0)$

**Important to know:**

According to the definition of an ellipse, if we take any point on the circumference of the ellipse and draw one chord to one focus and another chord to the other focus,

we will find that their sum is equal to $2a$

How do we find $c$?

According to the formula $a^2=b^2+c^2$

And now? For the real practice!

Here is the following ellipse equation:

$\frac{x^2}{16} +\frac{y^2}{25} =1$

Find $a$ and $b$

Solution:

If we look at the ellipse equation, we see that in the denominator $a$ and $b$ are squared.

Therefore, we need to take the square root of $16$ and the square root of $25$ to identify $a$ and $b$.

We get that:

$A = 4$

$B= 5$

**Another exercise:**

In front of you is an ellipse whose intersection points with the $X$ axis are $(-3,0)(0,3)$

and its intersection points with the $Y$ axis are $(0 , 6 )$ and $(0 , -6 )$

Find the equation of the ellipse

Solution:

We know that –

The intersection points of the ellipse with the $X$ axis are:

$(a,0)$ and – $(-a, 0)$

The intersection points of the ellipse with the $Y$ axis are:

$( 0, b)$ and $(0, -b)$

Therefore, if we substitute the given intersection points, we can immediately identify $a$ and $b$.

$a = 3$

$b = 6$

Now we substitute the $a$ and $b$ of the ellipse into the ellipse equation:

$\frac{x^2}{a^2} +\frac{y^2}{b^2} =1$

And we get that the equation of the given ellipse is:

$\frac{x^2}{3^2} +\frac{y^2}{6^2} =1$

$\frac{x^2}{9} +\frac{y^2}{36} =1$

To calculate the area of an ellipse, you should be familiar with two more concepts.

In an ellipse, there is a major radius – the vertical one

and a minor radius – the horizontal one

Let's see this in the illustration:

*Illustration in a Word file*

$A$ - The major radius is on the $Y$ axis and is marked in purple

$B$ - The minor radius is on the $X$ axis and is marked in pink

We will use the formula to calculate the area of an ellipse:

$S$ Area of an ellipse = $π*A*B$

**Note – **

If you find the intersection points of the ellipse with the $X$ axis and the $Y$ axis, you can find $A$ and $B$ which represent the distance of the ellipse from the axes and thus find the area of the ellipse.

**And now to practice!**

Here is the following ellipse equation:

$\frac{x^2}{9} +\frac{y^2}{36} =1$

Find \(a \) and \(b \).

Find the points of intersection with the \(X\) axis and the points of intersection with the \(Y\) axis.

Find the area of the ellipse.

Solution:

If we look at the ellipse equation, we see that in the denominator $a$ and $b$

are squared.

Therefore, we need to take the square root of $9$ and the square root of $64$ to identify $a$ and $b$

We get that:

$a = 3$

$b= 8$

It is known that:

The intersection points of the ellipse with the $X$ axis are:

$(a,0)$ and $(-a, 0)$

The intersection points of the ellipse with the $Y$ axis are:

$( 0, b)$ and $(0 ,-b)$

Therefore, we simply substitute the $a$ and $b$ we found and get that:

The intersection points of the ellipse with the $X$ axis are:

$(0,3)$ and $(-3, 0)$

The intersection points of the ellipse with the $Y$ axis are:

$( 0, 8)$ and $( 0 ,-8)$

To find the area of the ellipse, we need to find $a$ and $b$

In fact, we already found them when we found the intersection points:

$A = 8$ the distance from the center of the ellipse to the intersection with the $Y$ axis

$B = 3$ the distance from the center of the ellipse to the intersection with the $X$ axis

Substitute into the formula and get:

$π*8*3=75.36$

The area of the ellipse is $75.36$ cm²

Want to know more about the area of an ellipse? Click here