# Ellipse

Written by Jerry Ratzlaff on . Posted in Plane Geometry

## Ellipse

An ellipse is a flat plane curve that when add togeather any two distances from any point on the ellipse to each of the foci will always equal the same.

## Standard Ellipse

### Standard Ellipse Formula

$$\frac {x^2}{a^2} \;+\; \frac {y^2}{x^2} \;=\; 1$$

horizontal : $$\; \frac { \left( x - h \right )^2 } { a^2 } \;+\; \frac { \left( y - k \right )^2 } { b^2 } \;=\; 1$$

vertical : $$\; \frac { \left( x - h \right )^2 } { b^2 } \;+\; \frac { \left( y - k \right )^2 } { a^2 } \;=\; 1$$

Where:

$$x$$ = horizontal coordinate of a point on the ellipse

$$y$$ = vertical coordinate of a point on the ellipse

$$a$$ = length semi-minor axis

$$b$$ = length semi-minor axis

$$h$$ and $$k$$ = center point of ellipse

## Area of an Ellipse

### Area of an Ellipse formula

$$A = \pi a b$$          $$area \;=\; Pi \;\;x\;\; axis \; a \;\;x\;\; axis \; b$$

Where:

$$A$$ = area

$$a$$ = length semi-major axis

$$b$$ = length semi-minor axis

$$\pi$$ = Pi

## Foci of an Ellipse

Foci is a point used to define the conic section.  $$F$$ and $$G$$ seperately are called "focus", both togeather are called "foci".

### Foci of an Ellipse formula

$$c^2 = a^2 - b^2$$          $$center \; to \; focus^2 \;=\; semi-major \; axis^2 \;-\; semi-minor \; axis^2$$

Where:

$$c$$ = length center to focus

$$a$$ = length semi-major axis

$$b$$ = length semi-minor axis

$$F$$ and $$G$$ = focus

## Perimeter of an Ellipse

The 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".

### Perimeter of an Ellipse formula

$$p \approx 2\pi \sqrt { \frac{1}{2} \left(a^2 + b^2 \right) }$$          $$perimeter \;\approx\; 2 \;\;x\;\; Pi \; \sqrt { \; \frac{1}{2} \; \left( \; axis \; a^2 \;+\; axis \; b^2 \; \right) }$$

Where:

$$p$$ = perimeter

$$a$$ = length semi-major axis

$$b$$ = length semi-minor axis

$$\pi$$ = Pi

## Semi-major and Semi-minor Axis of an Ellipse

The major axis is always the longest axis in an ellipse.

The minor axis is always the shortest axis in an ellipse.

### Semi-major and Semi-minor Axis of an Ellipse formula

$$a = \frac{A}{\pi b}$$          $$axis \; a \;=\; \frac{ area }{ Pi \;\;x\;\; axis \; b}$$

$$a = \frac {l} {1- \epsilon^2}$$          $$axis \; a \;=\; \frac { semi-latus \; rectum } { 1 \;-\; eccentricity^2}$$

$$b = \frac{A}{\pi a}$$          $$axis \; b \;=\; \frac{ area }{ Pi \;\;x\;\; axis \; a}$$

$$b = a \sqrt {1- \epsilon^2}$$          $$axis \; b \;=\; semi-major \; axis \; \sqrt { \; 1 \;-\; eccentricity^2}$$

Where:

$$a$$ = length semi-major axis

$$b$$ = length semi-minor axis

$$A$$ = area

$$\pi$$ = Pi

$$l$$ = semi-latus rectum

$$\epsilon$$ (Greek symbol epsilon) = eccentricity