Ellipse Sector

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • ellipse sector 4Ellipse 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