Sector of an Ellipse

on . Posted in Plane Geometry

  • ellipse sector 4ellipse catenary curve 1Ellipse 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.

 

 

 

 

Sector Area formula

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

 

Sector Radius formula

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

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

Symbol 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 \(\large{ in }\) \(\large{ mm }\)

 

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