This equation of an ellipse calculator is a handy tool for determining the basic parameters and most important points on an ellipse. You can use it to find its center, vertices, foci, area, or perimeter. All you need to do is to write the ellipse standard form equation and watch this calculator do the math for you.
This article was written to help you understand the basic features of an ellipse. Read on to learn how to find the area of an oval, what is the focus of an ellipse, or how do you define the eccentricity.
If you like the ellipse calculator, try the octagon calculator, too!
What is an ellipse?
An ellipse is a generalized case of a closed conical section. It is oval in shape and is obtained if you slice a cone with an inclined plane. In the case when the inclination angle of the plane is equal to zero, you obtain a circle (circles are a subset of ellipses).
If you want to draw an ellipse, you need to determine two points, called foci (points F₁ and F₂ in the image above). Then, the ellipse is defined as a set of all points for which the sum of distances to the first and the second focus is equal to a constant value. In a circle, both foci overlap at one point.
Ellipse standard form
The equation of an ellipse is a generalized case of the equation of a circle. It has the following form:
(x - c₁)² / a² + (y - c₂)² / b² = 1
- (x, y) are the variables - the coordinates of an arbitrary point on the ellipse;
- (c₁, c₂) are the coordinates of the ellipse's center;
- a is the distance between the center and the ellipse's vertex, lying on the horizontal axis;
- b is the distance between the center and the ellipse's vertex, lying on the vertical axis.
If the ellipse is horizontal (i.e. it is a circle "stretched" along the horizontal axis), then a is greater than b. If it is vertical, then b is greater than a. For the parameters
a = b the ellipse is a regular circle of radius a and the following equation of a circle:
(x - c₁)² + (y - c₂)² = a²
How to find the area of an oval?
Once you know the equation of an ellipse, you can calculate its area. It is actually a very simple task. First, recall the formula for the area of a circle:
A = πr²
In the case of an ellipse, you don't have a single value for a radius, but two different values of a and b. All you have to do is to substitute their product in the place of r²:
A = π * a * b
Surprisingly, finding the perimeter of an ellipse is much harder. There are many approximations that give solutions at various precision and accuracy levels. Our ellipse calculator uses the approximation given by Ramanujan:
P = π * (a + b) * (1 + 3 * (a-b)²/(a+b)²) / (10 + √[4 - 3*(a-b)²/(a+b)²])
Eccentricity of an ellipse
Our ellipse standard form calculator can also provide you with the eccentricity of an ellipse. What is this value? It is a ratio of two values: the distance between any point of the ellipse and the focus, and the distance from this arbitrary point to a line called the directrix of the ellipse.
Every ellipse is characterized by a constant eccentricity. If the ellipse is a circle, then the eccentricity is 0. If it is infinitely close to a straight line, then the eccentricity approaches infinity.
Eccentricity is calculated with the use of the following equation:
eccentricity = √(a² - b²) / afor a horizontal ellipse, and
eccentricity = √(b² - a²) / bfor a vertical ellipse.
Center, foci, and vertices of an ellipse
Apart from the basic parameters, our ellipse calculator can easily find the coordinates of the most important points on every ellipse. These points are the center (point C), foci (F₁ and F₂), and vertices (V₁, V₂, V₃, V₄).
- To find the center, take a look at the equation of the ellipse. The coordinates of the center are simply the numbers (c₁, c₂).
- The foci of a horizontal ellipse are:
F₁ = (-√(a² - b²)+ c₁, c₂)
F₂ = (√(a² - b²)+ c₁, c₂)
- The foci of a vertical ellipse are:
F₁ = (c₁, -√(b² - a²)+ c₂)
F₂ = (c₁, √(b² - a²)+ c₂)
- Vertices of an ellipse are located at the points:
V₁ = (-a + c₁, c₂)
V₂ = (a + c₁, c₂)
V₃ = (c₁, -b + c₂)
V₄ = (c₁, b + c₂)