Segment of a Circle

Written by Jerry Ratzlaff 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.

 

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Arc Length of a Segment formula

\(\large{ L =   \Delta \; \frac{\pi}{180} \; r }\)   

Where:

 Units English Metric
\(\large{ L }\) = arc length \(\large{ in }\)  \(\large{ mm }\)
\(\large{ \Delta }\) = angle \(\large{ deg }\)   \(\large{ rad }\) 
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius \(\large{ in }\)   \(\large{ mm }\) 

 

area of a Segment formulas

\(\large{ A =   \frac {r^2} {2} \; \left( \Delta -  sin \; \Delta   \right)   }\)   
\(\large{ A =   \frac {r^2 \; \left( \Delta \;-\;  sin \; \Delta   \right) }{ 2 }       }\)  
\(\large{ A =  r^2 \; \left(  \frac { \Delta \; \pi }{ 360 }  \;-\; \frac { sin \; \Delta }{ 2 }  \right)  }\)   
\(\large{ A =  \frac { 1 }{ 2 } \; r^2 \; \left( \;  \frac {\pi}{180}  \Delta \;-\; sin \; \Delta \; \right)   \;\;  }\)   

Where:

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

 

Distance from Centroid of a Segment formulas

\(\large{ C_x =  0  }\)   
\(\large{ C_y =  \frac {4 \; r}{3} \; \left(  \frac {sin^3 \; \frac{\Delta}{2} } {\Delta \; - \; sin \; \Delta}    \right)  }\)   

Where:

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

 

Elastic Section Modulus of a Segment formula

\(\large{ S =  \frac{ I_x }{ C_y \;-\; r \; cos \; \left(  \frac {\Delta}{2}   \right)  }  }\)   

Where:

 Units English Metric
\(\large{ S }\) = elastic section modulus \(\large{ in^4 }\)   \(\large{ mm^4 }\) 
\(\large{ C_x, C_y }\) = distance from centroid \(\large{ in }\)  \(\large{ mm }\)
\(\large{ \Delta }\) = angle \(\large{ deg }\)   \(\large{ rad }\) 
\(\large{ I }\) = moment of inertia \(\large{ in^4 }\)   \(\large{ mm^4 }\) 
\(\large{ r }\) = radius \(\large{ in }\)   \(\large{ mm }\) 

 

Height of a Segment formulas

\(\large{ h =  r  \; \left( 1 - cos \; \frac{ \Delta }{2} \right)    }\)   
\(\large{ h =  r  -  \sqrt{ r^2 - \frac{ l^2 }{4} }   }\)   

Where:

 Units English Metric
\(\large{ h }\) = height \(\large{ in }\)  \(\large{ mm }\)
\(\large{ \Delta }\) = angle \(\large{ deg }\)   \(\large{ rad }\) 
\(\large{ l }\) = chord \(\large{ in }\)   \(\large{ mm }\) 
\(\large{ r }\) = radius \(\large{ in }\)   \(\large{ mm }\) 

 

Length of a Segment formulas

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

Where:

 Units English Metric
\(\large{ c }\) = chord \(\large{ in }\)   \(\large{ mm }\) 
\(\large{ \Delta }\) = angle \(\large{ deg }\)   \(\large{ rad }\) 
\(\large{ r }\) = radius \(\large{ in }\)   \(\large{ mm }\) 

 

Perimeter of a Segment formula

\(\large{ P = \Delta \; r   + 2r \; sin \;  \frac { \Delta }{2} }\)   

Where:

 Units English Metric
\(\large{ P }\) = perimeter \(\large{ in }\)   \(\large{ mm }\) 
\(\large{ \Delta }\) = angle \(\large{ deg }\)   \(\large{ rad }\) 
\(\large{ r }\) = radius \(\large{ in }\)   \(\large{ mm }\) 

 

Polar Moment of Inertia of a Segment formula

\(\large{ J_{z} =   \frac {r^4}{4} \; \left(  \Delta - sin \; \Delta + \frac  {2}{3}  \; sin \; \Delta \; sin^2 \; \frac {\Delta}{2}  \right)    }\)   

Where:

 Units English Metric
\(\large{ J }\) = torsional constant  \(\large{ in^4 }\)  \(\large{ mm^4 }\)
\(\large{ \Delta }\) = angle \(\large{ deg }\)  \(\large{ rad }\)
\(\large{ r }\) = radius \(\large{ in }\)  \(\large{ mm }\)

 

Radius of Gyration of a Segment formulas

\(\large{ k_{x} =    \sqrt {   \frac {I_x}{A}   }   }\)   
\(\large{ k_{y} =   \sqrt {   \frac {I_y}{A} }   }\)   
\(\large{ k_{z} =   \sqrt {   k_{x}{^2}  +  k_{y}{^2}   }        }\)   

Where:

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

 

Second Moment of Area of a Segment formulas

\(\large{ I_{x} =   \frac {r^4}{8} \;  \left(  \Delta - sin \;  \Delta +  2 \; sin \; \Delta \; sin^2 \; \frac {\Delta}{2}  \right)    }\)   
\(\large{ I_{y} =   \frac {r^4}{24} \; \left(  3 \; \Delta - 3 \; sin \;  \Delta -  2 \; sin \; \Delta \; sin^2 \; \frac {\Delta}{2}  \right)    }\)   

Where:

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

 

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Tags: Inertia Equations Structural Steel Equations Modulus Equations