Sector of a Circle

on . Posted in Plane Geometry

  • circle sector 5circle sector 11Sector is a fraction of the area of a circle with a radius on each side and an arc.
  • Angle (\(\Delta\))  -  Two rays sharing a common point.
  • Center (cp)  -  Having all points on the line circumference are at equal distance from the center point.
  • Chord (c)  -  Also called long chord (LC), is between any two points on a circular curve.
  • Circumference (C)  -  The outside of a circle or a complete circular arc.
  • Height (h)  -  Length of radius from radius center to midpoint of chord.
  • Height (h')  -  Length of radius from midpoint of chord to point on circular curve.
  • Length (L)  -  Total length of any circular curve measured along the arc.
  • Radius (r)  -  Half the diameter of a circle.  A line segment between the center point and a point on a circle or sphere.
  • Radius Point (rp)  -  Radius center point of circular curve.
  • Segment is an interior part of a circle bound by a chord and an arc.
  • Tangent (T)  -  A line that touches a curve at just one point such that it is perpendicular to a radius line of the curve.

Sector of a Circle Index

 

Angle of a Sector formula

\( \Delta \;=\;  2 \; A \;/\;r^2 \) 
Symbol English Metric
\( \Delta \) = angle \( deg \) \(rad \)
\( A \) = area of sector \( in^2 \) \(mm^2 \)
\( r \) = radius \( in \) \( mm \)

 

Arc Length of a Sector formula

\( L \;=\;   \Delta \; (\pi\;/\;180) \; r \) 
Symbol English Metric
\( L \) = arc length \( in \) \(mm \)
\( \Delta \) = angle \( deg \) \(rad \)
\( \pi \) = Pi \(3.141 592 653 ...\)
\( r \) = radius \( in \) \( mm \)

 

area of a Sector formula

\( A \;=\; \Delta \;r^2 \;/\;2 \) 

\( A \;=\; ( \Delta \;/\; 360 ) \; \pi \; r^2 \) 

\( A \;=\; ( \Delta \; \pi \;/\; 360 ) \; r^2  \) 

Symbol English Metric
\( A \) = area of sector \( in^2 \) \(mm^2 \)
\( \Delta \) = angle \( deg \) \(rad \)
\( \pi \) = Pi \(3.141 592 653 ...\)
\( r \) = radius \( in \) \( mm \)

 

Distance from Centroid of a Sector formulas

\( C_x \;=\; 2 \; r \; [\; sin(\Delta) \;/\;3\; \Delta \;]  \) 

\( C_y \;=\; 0  \) 

Symbol English Metric
\( C \) = distance from centroid \( in \) \( mm \)
\( A \) = area of sector \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \(rad \)
\( r \) = radius \( in \) \( mm \)

 

Elastic Section Modulus of a Sector formula

\( S \;=\; I_x \;/\; sin(\Delta) \; r \) 
Symbol English Metric
\( S \) = elastic section modulus \( in^3 \)  \( mm^3 \)
\( \Delta \) = angle \( deg \) \(rad \)
\( I \) = moment of inertia \( in^4 \)  \( mm^4 \)
\( r \) = radius \( in \) \( mm \)

 

Perimeter of a Sector formula

\( P \;=\;   2 \; r  +  2 \; r \; \Delta \) 
Symbol English Metric
\( P \) = perimeter \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \(rad \)
\( r \) = radius \( in \) \( mm \)

 

Polar Moment of Inertia of a Sector formulas

\( J_{z} \;=\;   (r^4\;/\;18)  \; ( 9 \; \Delta^2 - 8 \; sin^2(\Delta) \;/\; \Delta  )  \) 

\( J_{z1} \;=\;  r^4 \; \Delta\;/\;2  \) 

Symbol English Metric
\( J \) = torsional constant  \( in^4 \)  \( mm^4 \)
\( \Delta \) = angle \( deg \) \(rad \)
\( r \) = radius \( in \) \( mm \)

 

Radius of a Sector formula

\( r \;=\; \sqrt{ 2 \; A \;/\; \Delta  } \) 
Symbol English Metric
\( r \) = radius \( in \) \( mm \)
\( A \) = area of sector \( in^2 \) \(mm^2 \)
\( \Delta \) = angle \( deg \) \(rad \)

 

Radius of Gyration of a Sector formulas

\( k_{x} \;=\;   \frac{1}{4} \; \sqrt{  2 \; r^2 \; [\; 2\; \Delta - sin(2 \; \Delta ) \;/\; \Delta \;]  }  \) 

\( k_{y} \;=\; \frac{1}{12} \;  \sqrt{  2 \; r^2 \; [\; 180^2 + 9\; \Delta \; sin(2\; \Delta) - 32 + 32 \; cos^2(\Delta)  \;/\; \Delta^2 \;]    }   \) 

\( k_{z} \;=\; \frac{1}{6}  \; \sqrt{ 2 \; r^2  \; [\; 9 \; \Delta^2 - 8 \; sin^2(2\; \Delta ) \;/\; \Delta^2 \;] }  \)

\( k_{x1} \;=\; \frac{1}{4}  \; \sqrt{ 2 \; r^2 \; [\; 2\; \Delta - sin(2 \; \Delta) \;/\; \Delta \;]   }   \)

\( k_{y1} \;=\; \frac{1}{4}  \; \sqrt{ 2 \; r^2 \; [\; 2\; \Delta + sin(2 \; \Delta ) \;/\; \Delta \;]  }  \)

\( k_{x1} \;=\; r\;/\; \sqrt{2}   \)

Symbol English Metric
\( k \) = radius of gyration \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \(rad \)
\( r \) = radius \( in \) \( mm \)

 

Second Moment of Area of a Sector formulas

\( I_{x} \;=\; (r^4\;/\;4) \; [\; \Delta - \frac{1}{2} \; sin( 2 \; \Delta )  \;]   \) 

\( I_{y} \;=\;  (r^4\;/\;4) \; [\; \Delta + \frac{1}{2} \; sin( 2 \; \Delta)  \;] - [\; (4\;r^4\;/\;9\; \Delta ) \; sin^2 ( \Delta ) \;] \) 

\( I_{x1} \;=\; I_x  +  r^4 \; \Delta \; sin^2 (\Delta) \) 

\( I_{y1} \;=\; (r^4\;/\;4) \; [\; \Delta +  \frac{1}{2} \; sin( 2 \; \Delta )  \;]    \)

Symbol English Metric
\( I \) = moment of inertia  \( in^4 \)  \( mm^4 \)
\( \Delta \) = angle \( deg \) \(rad \)
\( r \) = radius \( in \) \( mm \)

 

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Tags: Structural Steel Circle