# Ellipse

Written by Jerry Ratzlaff on . Posted in Plane Geometry

•  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".
• The major axis is always the longest axis in an ellipse.
• The minor axis is always the shortest axis in an ellipse.

## Formulas that use Standard Ellipse

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

$$\large{ x }$$ = horizontal coordinate of a point on the ellipse

$$\large{ y }$$ = vertical coordinate of a point on the ellipse

$$\large{ a }$$ = length semi-major axis

$$\large{ b }$$ = length semi-minor axis

$$\large{ h }$$ and $$\large{ k }$$ = center point of ellipse

## Formulas that use Ellipse Area

 $$\large{ A = \pi \;a\; b }$$

### Where:

$$\large{ A }$$ = area

$$\large{ a }$$ = length semi-major axis

$$\large{ b }$$ = length semi-minor axis

$$\large{ \pi }$$ = Pi

## Formulas that use Ellipse Foci

 $$\large{ c^2 = a^2 - b^2 }$$

### Where:

$$\large{ c }$$ = length center to focus

$$\large{ a }$$ = length semi-major axis

$$\large{ b }$$ = length semi-minor axis

$$\large{ F }$$ and $$\large{ G }$$ = focus

## Formulas that use Ellipse Perimeter

This is an approximate perimeter of an ellipse formula.  There is no easy way to calculate the ellipse perimeter with high accuracy.

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

### Where:

$$\large{ p }$$ = perimeter approximation

$$\large{ a }$$ = length semi-major axis

$$\large{ b }$$ = length semi-minor axis

$$\large{ \pi }$$ = Pi

## Formulas that use Ellipse Semi-major Axis Length

 $$\large{ a = \frac{A}{\pi \; b} }$$

### Where:

$$\large{ a }$$ = length semi-major axis

$$\large{ A }$$ = area

$$\large{ b }$$ = length semi-minor axis

$$\large{ \pi }$$ = Pi

## Formulas that use Ellipse Semi-minor Axis Length

 $$\large{ b = \frac{A}{\pi \; a} }$$

### Where:

$$\large{ b }$$ = length semi-minor axis

$$\large{ A }$$ = area

$$\large{ a }$$ = length semi-major axis

$$\large{ \pi }$$ = Pi