Ellipse (a two-dimensional figure) is a conic section or a stretched circle. It is a flat plane curve that when adding togeather any two distances from any point on the ellipse to each of the foci will always equal the same.- Foci is a point used to define the conic section. F and G seperately are called "focus", both togeather are called "foci".
- The perimeter of an ellipse formula is an approximation that is about 5% of the true value as long as "a" is no more than 3 times longer than "b".
- Latus rectum is a line drawn perpencicular to the transverse axis of the ellipse and is passing through the foci of the ellipse.
- The major axis is always the longest axis in an ellipse.
- The minor axis is always the shortest axis in an ellipse.
Standard Ellipse formulas
| \(\large{ \frac {x^2}{a^2} + \frac {y^2}{x^2} = 1 }\) |
|
| \(\large{ \left( \frac {x}{a} \right)^2 + \left( \frac {y}{x} \right)^2 = 1 }\) |
|
| \(\large{ \frac { \left( x \;-\; h \right )^2 } { a^2 } + \frac { \left( y \;-\; k \right )^2 } { b^2 } = 1 }\) |
(major axis horizontal) |
| \(\large{ \frac { \left( x \;-\; h \right )^2 } { b^2 } + \frac { \left( y \;-\; k \right )^2 } { a^2 } = 1 }\) |
(major axis vertical) |
Where:
| Units |
English |
Metric |
| \(\large{ x }\) = horizontal coordinate of a point on the ellipse |
- |
- |
| \(\large{ y }\) = vertical coordinate of a point on the ellipse |
- |
- |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ h }\) and \(\large{ k }\) = center point of ellipse |
- |
- |
Ellipse Area formula
| \(\large{ A = \pi \;a\; b }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ A }\) = area |
\(\large{ in^2 }\) |
\(\large{ mm^2 }\) |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ \pi }\) = Pi |
\(\large{3.141 592 653 ...}\) |
Ellipse Circumference formula
| \(\large{ C = 2\;\pi \; \sqrt{ \frac{ a^2 \;+\; b^2 }{ 2 } } }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ C }\) = circumference |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ \pi }\) = Pi |
\(\large{3.141 592 653 ...}\) |
Ellipse eccentricity formula
| \(\large{ \epsilon = \sqrt{ \frac{ a^2 \;-\; b^2 }{ a^2 } } }\) |
|
| \(\large{ \epsilon = \left( \frac{ 1 \;-\; b^2 }{ a^2 } \right)^{0.5} }\) |
|
| \(\large{ \epsilon = \sqrt{ 1 - \frac{ b^2 }{ a^2 } } }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ \epsilon }\) (Greek symbol epsilon) = eccentricity |
\(\large{ dimensionless }\) |
| \(\large{ a }\) = one-half of the ellipse's major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = one-half of the ellipse's minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
Ellipse Foci formula
| \(\large{ c^2 = a^2 - b^2 }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ c }\) = length center to focus |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ F }\) and \(\large{ G }\) = focus |
\(\large{ in }\) |
\(\large{ mm }\) |
Ellipse Perimeter formula
This is an approximate perimeter of an ellipse formula. There is no easy way to calculate the ellipse perimeter with high accuracy.
| \(\large{ p \approx 2\; \pi\; \sqrt { \frac{1}{2}\; \left(a^2 + b^2 \right) } }\) |
|
| \(\large{ p \approx 2\; \pi\; \sqrt { \frac{a^2 \;+\; b^2}{2} } }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ p }\) = perimeter approximation |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ \pi }\) = Pi |
\(\large{3.141 592 653 ...}\) |
Ellipse Latus Rectum formula
| \(\large{ a = \frac{ 2 \; b^2 }{ a } }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ L }\) = Latus rectum |
- |
- |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
Ellipse Semi-major Axis Length formula
| \(\large{ a = \frac{A}{\pi \; b} }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ A }\) = area |
\(\large{ in^2 }\) |
\(\large{ mm^2 }\) |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ \pi }\) = Pi |
\(\large{3.141 592 653 ...}\) |
Ellipse Semi-minor Axis Length formula
| \(\large{ b = \frac{A}{\pi \; a} }\) |
|
Where:
| Units |
English |
Metric |
| \(\large{ A }\) = area |
\(\large{ in^2 }\) |
\(\large{ mm^2 }\) |
| \(\large{ a }\) = length semi-major axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ b }\) = length semi-minor axis |
\(\large{ in }\) |
\(\large{ mm }\) |
| \(\large{ \pi }\) = Pi |
\(\large{3.141 592 653 ...}\) |
