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Segment of a Circle

Arc Length of a Segment formula
\( L \;=\;   \Delta \cdot  \dfrac{ \pi }{ 180 } \cdot r \) 
Symbol English Metric
\( L \) = arc length \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( \pi \) = Pi \(3.141 592 653 ...\) \(3.141 592 653 ...\)
\( r \) = radius \( in \) \( mm \)

circle segment 11circle 17 

 

 

 

 

  • Segment is an interior part of a circle bound by a chord and an arc.
  • Angle (\(\Delta\))  -  Two rays sharing a common point.

Area of a Segment formula

\( A \;=\;   \dfrac{ r^2 }{2 } \cdot  (\; \Delta -  sin( \Delta) \;)   \) 

\( A \;=\;   \dfrac{  r^2 \cdot   (\; \Delta -  sin (\Delta) \; )  }{ 2  } \)

\( A \;=\; r^2 \cdot   \left(  \dfrac{ \Delta \cdot \pi }{ 360 }  -  \dfrac{ sin(\Delta) }{ 2 }  \right)  \) 

\( A \;=\; \dfrac{ 1 }{ 2 } \cdot r^2 \cdot  \left(   \dfrac{ \pi }{ 180 } \cdot \Delta - sin(\Delta) \right)   \) 
Symbol English Metric
\( A \) = area \( in^2 \)  \( mm^2 \)
\( \Delta \) = angle \( deg \) \( rad \)
\( \pi \) = Pi \(3.141 592 653 ...\) \(3.141 592 653 ...\)
\( r \) = radius \( in \) \( mm \)

circle segment 11

  • 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.
  • Circle  -  All points are at a fixed equal distance from a radius point (rp).
  • 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.

Distance from Centroid of a Segment formulas

\( C_x \;=\; 0 \)

\( C_y \;=\; \dfrac{ 4 \cdot r }{ 3 } \cdot  \dfrac{  sin^3 \dfrac{ \Delta }{ 2 }  }{ \Delta - sin(\Delta)  }  \)
Symbol English Metric
\( C_x, C_y \) = distance from centroid \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( r \) = radius \( in \) \( mm \)

circle segment 11

  • Major Arc  -  The longest of two arcs of a circle or ellipse.
  • Minor Arc  -  The shorter of two arcs of a circle or ellipse.
  • 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.
  • Sector is a fraction of the area of a circle with a radius on each side 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.
  • See Article Link  - Geometric Properties of Structural Shapes

Elastic Section Modulus of a Segment formula
\( S \;=\;  \dfrac{   I_x }{  C_y - r \cdot  cos \left( \dfrac{ \Delta }{ 2 } \right) }  \) 
Symbol English Metric
\( S \) = elastic section modulus \( in^3 \)   \( mm^3 \) 
\( C_x, C_y \) = distance from centroid \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( I \) = moment of inertia \(lbm\;/\;ft^2-sec\) \(kg\;/\;m^2\)
\( r \) = radius \( in \) \( mm \)

circle segment 11Piping Designer Logo 1

 

 

 

 

 

Height of a Segment formulas

\( h \;=\; r  \cdot   \left( 1 - cos \left(\dfrac{ \Delta }{ 2 } \right)  \right)  \) 

\( h \;=\; r  -  \sqrt{ r^2 -  \dfrac{ l^2 }{ 4 }  }  \) 
Symbol English Metric
\( h \) = height \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( l \) = chord \( in \) \( mm \)
\( r \) = radius \( in \) \( mm \)

circle segment 11

 

 

 

 

 

 

 

 

Length of a Segment formulas

\( c \;=\;  2 \cdot r \cdot  sin \left( \dfrac{ \Delta }{ 2 } \right)  \) 

\( c \;=\; r \cdot  \sqrt{ 2 - 2 \cdot cos ( \Delta ) }  \) 
Symbol English Metric
\( l \) = chord \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( r \) = radius \( in \) \( mm \)

circle segment 11

 

 

 

 

 

 

 

Perimeter of a Segment formula
\( P \;=\; \Delta \cdot  r  + 2 \cdot r \cdot  sin \left( \dfrac{ \Delta }{ 2 } \right)  \) 
Symbol English Metric
\( P \) = perimeter \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( r \) = radius \( in \) \( mm \)

 circle segment 11

 

 

 

 

 

Polar Moment of Inertia of a Segment formula
\( J_{z} \;=\;   \dfrac{r^4}{4}  \cdot  \left(  \Delta - sin( \Delta) + \dfrac{2}{3}  \cdot sin( \Delta) \cdot sin^2 \left( \dfrac{\Delta}{2} \right)   \right)   \) 
Symbol English Metric
\( J \) = torsional constant \( in^4 \) \( mm^4 \)
\( \Delta \) = angle \( deg \) \( rad \)
\( r \) = radius \( in \) \( mm \)

 circle segment 11