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 of an Ellipse Index

 

Area Sector of an Ellipse formula

\( A_{area} \;=\; \frac{a\;b}{2} \; \left( {\theta \;-\; atan\left[  \frac{  a\;-\;b \;sin(2\;\theta_1)  }{  a\;+\;b\;+\;\left(a\;-\;b\right)\;cos(2\;\theta_2)  } \right]  \;+\;  atan\left[  \frac{  a\;-\;b \;sin(2\;\theta_1)  }{  a\;+\;b\;+\;\left(a\;-\;b\right)\;cos(2\;\theta_2)  } \right]  }   \right)  \) 
Symbol English Metric
\( A_{area} \) = area \( in^2 \)  \( mm^2 \) 
\( \theta \) = angle \( deg\) \( rad \)
\( \theta_1 \) = angle \( deg \) \( rad \)
\( \theta_2 \) = angle \( deg \) \( rad \)
\( a \) = semi-major axis \( in \) \( mm \)
\( b \) = semi-minor axis \( in \) \( mm \)

 

Radius Sector of an Ellipse formula

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

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

Symbol English Metric
\( j \) = radius \( in \) \( mm \)
\( k \) = radius \( in \) \( mm \)
\( \theta_1 \) = angle \( deg \) \( rad \)
\( \theta_2 \) = angle \( deg \) \( rad \)
\( a \) = semi-major axis \( in \) \( mm \)
\( b \) = semi-minor axis \( in \) \( mm \)

 

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Tags: Ellipse