Segment of a Circle

on . Posted in Plane Geometry

  • circle 17circle segment 11Segment is an interior part of a circle bound by a chord 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.
  • 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.
  • 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.

Segment of a Circle Index

 

Arc Length of a Segment 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 Segment formula

\( A \;=\;   \frac {r^2} {2} \; [\; \Delta -  sin( \Delta) \;]   \) 

\( A \;=\;   r^2 \; [\; \Delta -  sin(\Delta) \;] \;/\; 2   \)

\( A \;=\; r^2 \; [\;  ( \Delta \; \pi \;/\; 360 )  - ( sin(\Delta) \;/\; 2 )  \;]  \) 

\( A \;=\; \frac{ 1 }{ 2 } \; r^2 \; [\; (\pi\;/\;180) \; \Delta - sin(\Delta) \;]   \) 

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

 

Distance from Centroid of a Segment formulas

\( C_x \;=\; 0 \)

\( C_y \;=\; \frac{4 \; r}{3} \; [\;  sin^3 ( \frac{\Delta}{2} ) \;/\; \Delta - sin(\Delta)  \;]  \)

Symbol English Metric
\( C_x, C_y \) = distance from centroid \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( r \) = radius \( in \) \( mm \)

 

Elastic Section Modulus of a Segment formula

\( S \;=\; I_x \;/\; C_y - r \; cos(  \frac {\Delta}{2} )  \) 
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 \( in \) \( mm \)

 

Height of a Segment formulas

\( h \;=\; r  \; [\; 1 - cos ( \frac{ \Delta }{2} ) \;]  \) 

\( h \;=\; r  -  \sqrt{ r^2 - ( l^2 \;/\;4 )  }  \) 

Symbol English Metric
\( h \) = height \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( l \) = chord \( in \) \( mm \)
\( r \) = radius \( in \) \( mm \)

 

Length of a Segment formulas

\( c \;=\; 2 \; r \; sin (\frac{ \Delta }{2} ) \) 

\( c \;=\; r \;  \sqrt{ 2 - 2 \; cos( \Delta) }  \) 

Symbol English Metric
\( l \) = chord \( in \) \( mm \)
\( \Delta \) = angle \( deg \) \( rad \)
\( r \) = radius \( in \) \( mm \)

 

Perimeter of a Segment formula

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

 

Polar Moment of Inertia of a Segment formula

\( J_{z} \;=\;   \frac{r^4}{4} \; [\; \Delta - sin( \Delta) + \frac{2}{3}  \; sin( \Delta) \; sin^2(\frac{\Delta}{2}) \;]   \) 
Symbol English Metric
\( J \) = torsional constant \( in^4 \) \( mm^4 \)
\( \Delta \) = angle \( deg \) \( rad \)
\( r \) = radius \( in \) \( mm \)

 

Radius of Gyration of a Segment formulas

\( k_{x} \;=\;  \sqrt{ I_x\;/\;A  }  \) 

\( k_{y} \;=\;  \sqrt{ I_y\;/\;A }  \) 

\( k_{z} \;=\;  \sqrt{  k_{x}{^2}  +  k_{y}{^2} }  \) 

Symbol English Metric
\( k \) = radius of gyration \( in \) \( mm \)
\( A \) = area \( deg \) \( rad \)
\( I \) = moment of inertia \( in^4 \)  \( mm^4 \)

 

Second Moment of Area of a Segment formulas

\( I_{x} \;=\;   \frac {r^4}{8} \; [\; \Delta - sin( \Delta) + 2 \; sin( \Delta) \; sin^2( \frac {\Delta}{2} ) \;]    \) 

\( I_{y} \;=\;   \frac {r^4}{24} \;  [\; 3 \; \Delta - 3 \; sin( \Delta) - 2 \; sin( \Delta) \; sin^2( \frac {\Delta}{2}) \;]   \) 

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