# Ellipse

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".
• 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

$$(x^2\;/\;a^2) + (y^2\;/\;x^2) \;=\; 1$$

$$\left( x\;/\;a \right)^2 + \left( y\;/\;x \right)^2 \;=\; 1$$

$$[\; \left( x - h \right )^2 \;/\; a^2 \;] + [ \; \left( y - k \right )^2 \;/\; b^2 \; ] \;=\; 1$$     (major axis horizontal)

$$[\; \left( x \;-\; h \right )^2 \;/\; b^2 \; ] + [\; \left( y \;-\; k \right )^2 \;/\; a^2 \;] \;=\; 1$$     (major axis vertical)

Symbol English Metric
$$x$$ = horizontal coordinate of a point on the ellipse $$in$$ $$mm$$
$$y$$ = vertical coordinate of a point on the ellipse $$in$$ $$mm$$
$$a$$ = length semi-major axis $$in$$ $$mm$$
$$b$$ = length semi-minor axis $$in$$ $$mm$$
$$h$$ and $$\large{ k }$$ = center point of ellipse $$in$$ $$mm$$

### Area of an Ellipse formula

$$A \;=\; \pi \;a\; b$$
Symbol English Metric
$$A$$ = area $$in^2$$ $$mm^2$$
$$a$$ = length semi-major axis $$in$$ $$mm$$
$$b$$ = length semi-minor axis $$in$$ $$mm$$
$$\pi$$ = Pi $$3.141 592 653 ...$$

### Circumference of an Ellipse formula

$$C \;=\; 2\;\pi \; \sqrt{ a^2 + b^2 \;/\; 2 }$$
Symbol English Metric
$$C$$ = circumference $$in$$ $$mm$$
$$a$$ = length semi-major axis $$in$$ $$mm$$
$$b$$ = length semi-minor axis $$in$$ $$mm$$
$$\pi$$ = Pi $$3.141 592 653 ...$$

### eccentricity of an Ellipse formula

$$\epsilon \;=\; \sqrt{ a^2 - b^2 \;/\; a^2 }$$

$$\epsilon \;=\; \left( 1 - b^2 \;/\; a^2 \right)^{0.5}$$

$$\epsilon \;=\; \sqrt{ 1 - ( b^2 \;/\; a^2 ) }$$

Symbol English Metric
$$\epsilon$$  (Greek symbol epsilon) = eccentricity $$dimensionless$$
$$a$$ = one half of the ellipse's major axis $$in$$ $$mm$$
$$a$$ = one half of the ellipse's minor axis $$in$$ $$mm$$

### Perimeter of an Ellipse formulaS

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

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

$$p \;\approx\; 2\; \pi\; \sqrt{ a^2 + b^2\;/\;2}$$

Symbol English Metric
$$p$$ = perimeter approximation $$in$$ $$mm$$
$$a$$ = length semi-major axis $$in$$ $$mm$$
$$b$$ = length semi-minor axis $$in$$ $$mm$$
$$\pi$$ = Pi $$3.141 592 653 ...$$

### Latus Rectum of an Ellipse formula

$$L \;=\; 2 \; b^2 \;/\; a$$
Symbol English Metric
$$L$$ = Latus rectum $$in$$ $$mm$$
$$a$$ = length semi-major axis $$in$$ $$mm$$
$$b$$ = length semi-minor axis $$in$$ $$mm$$

### Semi-major Axis Length of an Ellipse formula

$$a \;=\; A\;/\;pi \; b$$
Symbol English Metric
$$A$$ = area $$in^2$$ $$mm^2$$
$$a$$ = length semi-major axis $$in$$ $$mm$$
$$b$$ = length semi-minor axis $$in$$ $$mm$$
$$\pi$$ = Pi $$3.141 592 653 ...$$

### Semi-minor Axis Length of an Ellipse formula

$$b \;=\; A\;/\;pi \; a$$
Symbol English Metric
$$A$$ = area $$in^2$$ $$mm^2$$
$$a$$ = length semi-major axis $$in$$ $$mm$$
$$b$$ = length semi-minor axis $$in$$ $$mm$$
$$\pi$$ = Pi $$3.141 592 653 ...$$

Tags: Ellipse