# Sector of an Ellipse

Written by Jerry Ratzlaff on . Posted in Plane Geometry

•  Ellipse sector (a two-dimensional figure) is a part of the interior of an ellipse having two radius boundries and an arc.
• Sector is a fraction of the area of a ellipse with a radius on each side and an edge.
• Major axis is always the longest axis in an ellipse.
• Minor axis is always the shortest axis in an ellipse.
• Semi-major axis is half of the longest axis of an ellipse.
• Semi-minor axis is half of the shortest axis of an ellipse.

## Ellipse Sector Area formula

 $$\large{ A_{area} = \frac{a\;b}{2} \; \left( {\theta \;-\; atan\;\left[ \frac{ a\;-\;b \;sin\;\left(2\;\theta_1\right) }{ a\;+\;b\;+\;\left(a\;-\;b\right)\;cos\left(2\;\theta_2\right) } \right] \;+\; atan\;\left[ \frac{ a\;-\;b \;sin\;\left(2\;\theta_1\right) }{ a\;+\;b\;+\;\left(a\;-\;b\right)\;cos\;\left(2\;\theta_2\right) } \right] } \right) }$$

### Where:

 Units English Metric $$\large{ A_{area} }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$ $$\large{ \theta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ \theta_1 }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ \theta_2 }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ a }$$ = semi-major axis $$\large{ in }$$ $$\large{ mm }$$ $$\large{ b }$$ = semi-minor axis $$\large{ in }$$ $$\large{ mm }$$

## Ellipse Sector Radius formulas

 $$\large{ j = \sqrt{ \frac{a^2\;b^2}{a^2\;sin^2 \;\theta_1 \;+\; b^2\;cos^2 \;\theta_2} } }$$ $$\large{ k = \sqrt{ \frac{a^2\;b^2}{a^2\;sin^2 \; \theta_2 \;+\; b^2\;cos^2\;\theta_1} } }$$

### Where:

 Units English Metric $$\large{ j }$$ = radius $$\large{ in }$$ $$\large{ mm }$$ $$\large{ k }$$ = radius $$\large{ in }$$ $$\large{ mm }$$ $$\large{ \theta_1 }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ \theta_2 }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ a }$$ = semi-major axis $$\large{ in }$$ $$\large{ mm }$$ $$\large{ b }$$ = semi-minor axis $$\large{ in }$$ $$\large{ mm }$$ 