Hollow Circle

on . Posted in Plane Geometry

  • Two circles each having all points on each circle at a fixed equal distance from a center point.
  • Center of a circle having all points on the line circumference are at equal distance from the center point.
  • A hollow circle is a structural shape used in construction.
  • See Article Link  -  Geometric Properties of Structural Shapes

Hollow Circle Index

 

circle hollow 4

area of a Hollow Circle formula

\(\large{ A =   \pi \; \left(  R^2  - r^2  \right)    }\)     (Area of a Hollow Circle)

\(\large{ R =  \sqrt{  \frac{ A \;-\; \pi  \; r^2  }{ \pi }   }  }\) 

\(\large{ r =  \sqrt{  R^2 - \frac{ A}{\pi}   }  }\) 

Solve for A

outside radius, R
inside radius, r

Solve for R

 
area of a hollow circle, A
inside radius, r

Solve for r

area of a hollow circle, A
outside radius, R

Symbol English Metric
\(\large{ A }\) = area  \(\large{ in^2 }\) \(\large{ mm^2 }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)

  

circle hollow 4

Circumference of a Hollow Circle formulas

\(\large{ C =   2 \; \pi \; r   }\)     (inside) 

\(\large{ C =  2 \; \pi \; R   }\)     (outside)

Solve for C

radius, r

Solve for r

circumference, C

Symbol English Metric
\(\large{ C }\) = circumference \(\large{ in }\) \(\large{ mm }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)

  

circle hollow 4

Distance from Centroid of a Hollow Circle formulas

\(\large{ C_x =  r}\)

\(\large{ C_y =  r}\)

Symbol English Metric
\(\large{ C_x, C_y }\) = distance from centroid \(\large{ in }\) \(\large{ mm }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)

 

circle hollow 4

Elastic Section Modulus of a Hollow Circle formula

\(\large{ S =      \frac{  \pi \; \left(  R^4  \;-\; r^4  \right) }{ 4\;R }     }\) 
Symbol English Metric
\(\large{ S }\) = elastic section modulus \(\large{in^3}\) \(\large{mm^3}\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)

 

circle hollow 4

Plastic Section Modulus of a Hollow Circle formula

\(\large{ Z =       \frac   { 4 \; \left(  R^3  \;-\; r^3  \right) }  { 3 }    }\) 
Symbol English Metric
\(\large{ Z }\) = plastic section modulus \(\large{ in^3 }\) \(\large{mm^3  }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)

 

circle hollow 4

Polar Moment of Inertia of a Hollow Circle formulas

\(\large{ J_{z} =  \frac { \pi }{2}  \;  \left(  R^4  - r^4  \right)   }\) 

\(\large{ J_{z1} =  \frac { \pi }{2}  \;  \left(  R^4  - r^4  \right)  +     2\; \pi \; R^2    \left(  R^2  - r^2  \right)     }\)

Symbol English Metric
\(\large{ J }\) = torsional constant \(\large{ in^4 }\) \(\large{mm^4  }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)

 

circle hollow 4

Radius of a Hollow Circle formula

\(\large{ r = \frac{C}{2 \; \pi} }\) 

\(\large{ R = \frac{C}{2 \; \pi} }\) 

Solve for r

circumference, C

Solve for C

radius, r

Symbol English Metric
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ C }\) = circumference \(\large{ in }\) \(\large{mm  }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)

 

circle hollow 4

Radius of Gyration of a Hollow Circle formulas

\(\large{ k_{x} =    \frac {1}{2}  \;  \sqrt {   R^2  + r^2   }     }\) 

\(\large{ k_{y} =   \frac {1}{2}  \;  \sqrt {   R^2  + r^2   }       }\) 

\(\large{ k_{z} =   \frac { \sqrt { 2 } }{2}  \;  \sqrt {   R^2  + r^2   }     }\) 

\(\large{ k_{x1} =  \frac {1}{2}  \;  \sqrt {  5 \; R^2  + r^2   }    }\)

\(\large{ k_{y1} =  \frac {1}{2}  \;  \sqrt {   5 \; R^2  + r^2   }   }\)

\(\large{ k_{z1} =   \frac { \sqrt { 2 } }{2}  \;  \sqrt {  5 \; R^2  + r^2   }   }\)

Symbol English Metric
\(\large{ k }\) = radius of gyration \(\large{ in }\) \(\large{mm  }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)

 

circle hollow 4

Second Moment of Area of a Hollow circle formulas

\(\large{ I_{x} =   \frac { \pi }{4} \;   \left(  R^4  - r^4  \right)     }\) 

\(\large{ I_{y} = \frac { \pi }{4} \;   \left(  R^4  - r^4  \right)    }\) 

\(\large{ I_{x1} =   \frac { \pi }{4}  \;  \left(  R^4  - r^4  \right)  +     \pi \; R^2    \left(  R^2  - r^2  \right) }\) 

\(\large{ I_{y1} =  \frac { \pi }{4}  \;  \left(  R^4  - r^4  \right)  +     \pi \; R^2    \left(  R^2  - r^2  \right) }\)

Symbol English Metric
\(\large{ I }\) = moment of inertia \(\large{ in^4 }\) \(\large{ mm^4 }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)

 

circle hollow 4

Sector of a Hollow Circle formula

\(\large{ A =  \frac{\pi \; \Delta \; \left( r^2 \;-\; R^2 \right)  }{360}  }\) 
Symbol English Metric
\(\large{ A }\) = sector area \(\large{ in^2 }\) \(\large{mm^2  }\)
\(\large{ \Delta }\) = angle \(\large{ deg }\) \(\large{rad  }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)

 

circle hollow 4

Torsional Constant of a Hollow Circle formulas

\(\large{ J  =  \frac {  \pi \; \left( R^4 \;-\;  r^4 \right)  } {  2  }   }\) 

\(\large{ J  =  \frac {  \pi \; \left( D^4 \;-\;  d^4 \right)  } {  32  }   }\) 

Symbol English Metric
\(\large{ J }\) = torsional constant \(\large{ in^4 }\) \(\large{mm^4  }\)
\(\large{ d }\) =  inside diameter \(\large{ in }\) \(\large{mm  }\)
\(\large{ D }\) =  outside diameter \(\large{ in }\) \(\large{mm  }\)
\(\large{ r }\) = inside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{mm  }\)
\(\large{ \pi }\) = Pi \(\large{3.141 592 653 ...}\)

 

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Tags: Inertia Structural Steel Modulus