What is Ellipse? List of Basic Ellipse Formulas

What is Ellipse?

The Ellipse in mathematics is a curve in a place surrounded by two focal points where the sum of distances between two focal points is always constant. Ellipse is the generalization of a circle or we can call it as the special type of Ellipse containing two focal points at similar locations. Is it possible to define the shape of Ellipse? Yes, it is possible and it is done through eccentricity whose value lies between 0 and 1.

If you would analyze analytically then Ellipse is a set of points that is defined as the ratio of the distance from any particular point to the curve whose sum is a constant number. This ratio in mathematical terms is popular as eccentricity. So, Ellipse or circle has the same definition or they are different? In case of a circle, the plane is parallel to the base of the come but in case of Ellipse, this is not parallel. So, they are different from each other.

Ellipse was discovered and studied first by Menaechmus. It was Euclid who wrote about Ellipse formula and equations. In 1602, scientists told that Mars was oval and earth is round based on Ellipse concept and the focal point. In 1705, there were some interesting facts discovered about Sun like it has elliptical orbit or its shape etc.

List of Basic Ellipse Formula

The Ellipse is the conic section that is closed and formed by the intersection of a cone by plane. They can be named as hyperbola or parabola and there are special formulas or equation to solve the tough Ellipse problems. Ellipse is the cross-section of a cylinder and parallel to the axis of the cylinder. With the help of basic Ellipse formulas, a lot of complex problems around the universe were possible to solve quickly.

\[\large Area\;of\;the\;Ellipse=\pi r_{1}r_{2}\]

\[\large Perimeter\;of\;the\;Ellipse=2\pi \sqrt{\frac{r_{1}^{2}+r_{2}^{2}}{2}}\]

Where,
r1 is the semi major axis of the ellipse.
r2 is the semi minor axis of the ellipse.

Question 1: Find the area and perimeter of a ellipse whose semi major axis is 10 cm and semi minor axis is 5 cm ?
Solution:

Given,
Semi major axis of the ellipse = r1 = 10 cm
Semi minor axis of the ellipse = r2 = 5 cm
Area of the ellipse
Area of the ellipse = πr1r2

Area of the ellipse = 3.14 X 10 X 5
Area of the ellipse = 157 cm2
Perimeter of the ellipse
\[\large =2\pi \sqrt{\frac{r_{1}^{2}+r_{2}^{2}}{2}}\]

\[\large =2\pi \sqrt{\frac{10^{2}+5^{2}}{2}}\]

\[\large =2\pi \sqrt{\frac{100+25}{2}}\]

\[\large =2\pi \sqrt{\frac{125}{2}}\]
Perimeter of the ellipse = 49.64 cm

Ellipse Equations

Ellipse is a set of points where two focal points together are named as Foci and with the help of those points, Ellipse can be defined. There are special equations in mathematics where you need to put Ellipse formulas and calculate the focal points to derive an equation. The sum of two focal points would always be a constant. Here is an example of the figure for clear understanding, what we meant by Ellipse and focal points exactly.

Ellipse Formulas

Why do students need to learn Ellipse Formula?

Ellipse is used widely for Physics and Engineering. It is used to calculate the orbit of solar system inside a planet based on the concept of focal points. The same concept is valid for the moon orbiting planets having two astronomical bodies. If you remember, the shape of stars or planets is also defined by ellipsoids.

Ellipse is the circle image under parallel projection and bounded case of perspective projection which is a simple intersection of the cone within the plane of projection. It can also be presented on the graph at horizontal or vertical axis with similar frequency. Further, it is used to understand the polarization effect in optical physics.


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