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Ellipse Calculator

(x-c₁)²/a² + (y-c₂)²/b² = 1
a
b
Eccentricity
Area
Perimeter
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The ellipse calculator finds the area, perimeter, and eccentricity of an ellipse.

By simply entering a few values into the calculator, it will nearly instantly calculate the eccentricity, area, and perimeter.

What is an Ellipse?

An ellipse is in the shape of an oval and many see it is a circle that has been squashed either horizontally or vertically. In fact the equation of an ellipse is very similar to that of a circle.  It is what is formed when you take a cone and slice through it at an angle that is neither horizontal or vertical. The section that is formed is an ellipse.

Ellipse Equation in Standard Form

Notice at the top of the calculator you see the equation in standard form, which is

\(\frac{(x – c_{1})^{2}}{a^{2}}+\frac{(y – c_{2})^{2}}{b^{2}}= 1\)

(x, y) are the coordinates of a point on the ellipse.

(\(c_{1}\), \(c_{2}\)) defines the coordinate of the center of the ellipse.

a is the horizontal distance between the center and one vertex

b is the vertical distance between the center and one vertex.

Note that if the ellipse is elongated vertically, then the value of b is greater than a. This makes sense because b is associated with vertical values along the y-axis. Similarly, if the ellipse is elongated horizontally, then a is larger than b. Remember, a is associated with horizontal values along the x-axis.

Finding the area of an ellipse may appear to be daunting, but it’s not too difficult once the equation is known. Notice that the formula is quite similar to that of the area of a circle, which is A = πr². Instead of r, the ellipse has a and b, representing distance from center to vertex in both the vertical and horizontal directions. Therefore, A = π • ab

While finding the perimeter of a polygon is generally much simpler than the area, that isn’t the case with an ellipse.  The calculator uses this formula

\(P = \pi\times (a+b)\times \frac{(1 + 3\times \frac{(a – b)^{2}}{(a+b)^{2}})}{10+\sqrt{((4 -3)\times (a + b)^{2})}}\)

 

Finally, the calculator will give the value of the ellipse’s eccentricity, which is a ratio of two values and determines how circular the ellipse is. The eccentricity value is always between 0 and 1. If you get a value closer to 0, then your ellipse is more circular. If you get a value closer to 1 then your ellipse is more oblong shaped.

The formula for eccentricity is as follows:

eccentricity = \(\frac{\sqrt{a^{2}-b^{2}}}{a}\) (horizontal)

eccentricity = \(\frac{\sqrt{b^{2}-a^{2}}}{b}\) (vertical)

You can see that calculating some of this manually, particularly perimeter and eccentricity is a bit time consuming. So give the calculator a try to avoid all this extra work.  You will be pleased by the accuracy and lightning speed that our calculator provides.