Sector of a Circle

Written by Jerry Ratzlaff 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.

 

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Angle of a Sector formula

\(\large{ \Delta = \frac{ 2 \; A }{r^2}  }\)   

Where:

 Units English Metric
\(\large{ \Delta }\) = angle \(\large{ deg }\) \(\large{ rad }\)
\(\large{ A }\) = area of sector \(\large{ in^2 }\) \(\large{ mm^2 }\)
\(\large{ r }\) = radius \(\large{ in }\) \(\large{ mm }\)

 

Arc Length of a Sector 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 Sector formula

\(\large{ A = \frac{ \Delta \;r^2 }{2} }\)   
\(\large{ A =  \frac { \Delta } { 360 } \; \pi \; r^2  \;\;  }\)   
\(\large{ A =  \frac { \Delta \; \pi } { 360 } \; r^2  \;\;  }\)   

Where:

 Units English Metric
\(\large{ A }\) = area of sector \(\large{ in^2 }\)  \(\large{ mm^2 }\)
\(\large{ \Delta }\) = 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 Sector formulas

\(\large{ C_x =  2 \; r \; \frac{sin \; \Delta}{3\; \theta}  }\)   
\(\large{ C_y =  0  }\)   

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 Sector formula

\(\large{ S =  \frac{ I_x }{ sin \;  \Delta \; r  }  }\)   

Where:

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

 

Perimeter of a Sector formula

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

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 Sector formulas

\(\large{ J_{z} =   \frac {r^4}{18}  \; \left(   \frac  {9 \; \Delta^2 \;-\; 8 \; sin^2 \; \Delta }{\Delta}   \right)    }\)   
\(\large{ J_{z1} =   \frac {r^4 \; \Delta}{2}     }\)   

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 a Sector formula

\(\large{ r = \sqrt{  \frac{ 2 \; A }{\Delta}  } }\)   

Where:

 Units English Metric
\(\large{ r }\) = radius \(\large{ in }\) \(\large{ mm }\)
\(\large{ A }\) = area of sector \(\large{ in^2 }\) \(\large{ mm^2 }\)
\(\large{ \Delta }\) = angle \(\large{ deg }\) \(\large{ rad }\)

 

Radius of Gyration of a Sector formulas

\(\large{ k_{x} =   \frac{1}{4} \;  \sqrt {  2 \; r^2 \; \frac{2 \; \theta \;-\; sin \; \left(2 \; \theta \right) }{\theta}     }   }\)   
\(\large{ k_{y} =  \frac{1}{12} \;  \sqrt {  2 \; r^2 \; \frac{180^2 \; + \; 9 \; \theta \; sin \; \left(2 \; \theta \right) \;-\; 32 \; + \; 32 \; cos^2 \; \theta    }{\theta^2}     }       }\)   
\(\large{ k_{z} =  \frac{1}{6}  \; \sqrt {  2 \; r^2  \; \frac{9 \; \Delta^2 \;-\; 8 \; sin^2 \; \left(2\; \Delta \right) }{\Delta^2}     }       }\)   
\(\large{ k_{x1} =  \frac{1}{4}  \; \sqrt {  2 \; r^2 \; \frac{2 \; \Delta \;-\; sin  \; \left(2 \; \Delta \right) }{\Delta}     }        }\)  
\(\large{ k_{y1} =  \frac{1}{4}  \; \sqrt {  2 \; r^2 \; \frac{2 \; \Delta \; + \; sin \; \left(2 \; \Delta \right) }{\Delta}     }        }\)  
\(\large{ k_{x1} =  \frac{r}{ \sqrt{2} }      }\)  

Where:

 Units English Metric
\(\large{ k }\) = radius of gyration \(\large{ in }\)   \(\large{ mm }\)
\(\large{ \Delta }\) = angle \(\large{ deg }\)  \(\large{ rad }\)
\(\large{ r }\) = radius \(\large{ in }\)  \(\large{ mm }\) 

 

Second Moment of Area of a Sector formulas

\(\large{ I_{x} =  \frac{r^4}{4} \; \left[ \Delta \;-\;  \frac{1}{2} \; sin \left( 2 \; \Delta \right)    \right]   }\)   
\(\large{ I_{y} =   \frac{r^4}{4} \; \left[ \Delta +  \frac{1}{2} \; sin \left( 2 \; \Delta \right)    \right]   \;-\;  \frac{4r^4}{9 \Delta} \; sin^2  \; \Delta   }\)   
\(\large{ I_{x1} =  I_x  +  r^4 \; \Delta \; sin^2 \;  \Delta }\)   
\(\large{ I_{y1} =  \frac{r^4}{4}  \left[ \Delta +  \frac{1}{2} \; sin \; \left( 2 \; \Delta \right)    \right]    }\)  

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