An ellipse is the set of all points in a plane such that the sum of their distances from two fixed points is constant. The fixed points are known as the foci, which are surrounded by the curve. Other important elements of ellipses are vertices, minor axis, major axis, center, and eccentricity. The shape of the ellipse is an oval and its area is defined by the length of the semi-minor axis and the length of the semi-major axis.
Here, we will learn more details of the elements of the ellipse along with diagrams that will help us to illustrate the concepts. Also, we will learn to find the area of an ellipse.
Definition of an ellipse
An ellipse is defined as the set of all points in the Cartesian plane, which have a distance from two fixed points (known as foci) that add up to form a constant value.
The ellipse is also defined as a conic section that is formed when a plane intersects a cone. The plane has to be inclined at an angle with respect to the base of the cone because if the plane is parallel to the base, the figure formed by the intersection is a circle.
Fundamental elements of an ellipse
The following are the fundamental elements of an ellipse:
- Major axis
- Minor axis
- Focal length
- Semi-minor axis
- Semi-major axis
The foci are the fixed points of the ellipse, which are located on the major axis. The foci are used to define the ellipse. Usually, we use the F to denote the foci.
The major axis is the longest diameter of the ellipse. The axes extend from one side of the ellipse to the other side and pass through the center. The total distance from one focus to any point on the ellipse plus the distance from that point to the other focus equals the length of the major axis.
The minor axis is the shortest diameter of the ellipse. We can also define the minor axis as the bisector (the line that divides another into two equal parts) perpendicular to the major axis.
The center of the ellipse is the point of intersection of the minor and major axes. We can define it as the center of symmetry of the ellipse.
Focal length is the length from one focus to the other.
The vertices are the points of intersection of the ellipse with the major axis. The vertices are the extreme points of the major axis.
The vertices are the intersection points of the ellipse with the minor axis. We can also define the covertices as the extreme points of the minor axis.
The semi-major axis is half of the major axis. The semi-major axis is the segment that goes from the center of the ellipse to a vertex of the ellipse and passes through one of the foci.
The semi-minor axis is half of the minor axis. The semi-minor axis is the segment that is perpendicular to the semi-major axis and runs from the center to a covertice.
Area of an ellipse
The following formula can be used to find the area of an ellipse:
Area = πab
- a is the length of the semi-major axis
- b is the length of the semi-minor axis
Interested in learning more about equations of an ellipse? Take a look at these pages: