Ellipse Formula
An ellipse is a locus of all the points in a plane, in a way that, the sum of their distances from the two fixed points, also called focus (singular is foci) in the plain, is constant. These focuses are surrounded by the curve. The constant ratio here is called the eccentricity of the ellipse and the fixed line is directrix. Eccentricity is a factor of the ellipse, and it further demonstrates the elongation of the Ellipse. In the next section, we will look at some ellipse formulas.
What Are Ellipse Formulas?
Perimeter of ellipse = \(2π\sqrt{\dfrac{a^2+b^2}{2}}\)
Area of ellipse = πab
e = \( \dfrac ca = \sqrt{1 \dfrac{b^2}{a^2} }\)
L = 2 b ^{2} /a
Equation of the ellipse with centre at (0,0) :
x ^{2} /a ^{2 } + y ^{2}/ b ^{2 } = 1
Equation of the ellipse with centre at (h,k) :
(xh)^{2} /a^{2} + (yk)^{2}/ b^{2} =1
where:
a > b
2a = length of major axis
a = semimajor axis
2b = length of minor axis
b = semiminor axis
e = eccentricity
L = length of latus rectum

Example 1:
Find the area of an ellipse whose major and minor axes are 14 in and 8 in respectively.
Solution:
To find: Area of an ellipse
Given: 2a = 14 in
a = 14/2 = 7
2b = 8 in
b = 8/2 = 4
Now, applying the ellipse formula for area:
Area of ellipse = π(a)(b)
= π(7)(4)
= 28π
= 28(22/7)
= 88 in^{2}
Answer: Area of the ellipse = 88 in^{2}.

Example 2:
The length of the semimajor and semiminor axis of an ellipse is 5 in and 3 in respectively. Find its eccentricity and the length of the latus rectum.
Solution:
To find: Eccentricity and the length of the latus rectum of an ellipse.
Given: a = 5 in, and b = 3 in
Now, applying ellipse formula for eccentricity:
\(e = \sqrt{1 \dfrac{ b^ 2}{ a^ 2} }\)
= √( 1  3^{2} / 5^{2} )
= √(1  9/25 )
= √((25  9)/25)
= √(16/25)
= 4/5
= 0.8
Now, applying ellipse formula for latus rectum:
L = 2 b ^{2} /a
= 2(3^{2})/5
= 2(9)/5
= 18/5
= 3.6 cm
Answer: Eccentricity and the length of the latus rectum of the ellipse are 0.8 and 3.6 in respectively.