# Ellipse Sector

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.

### Area of an Ellipse Sector 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:

$$\large{ A_{area} }$$ = area

$$\large{ \theta }$$ = angle

$$\large{ \theta_1 }$$ = angle

$$\large{ \theta_2 }$$ = angle

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

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

### Radius of an Ellipse Sector formula

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

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

Where:

$$\large{ j }$$ = radius

$$\large{ k }$$ = radius

$$\large{ \theta_1 }$$ = angle

$$\large{ \theta_2 }$$ = angle

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

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