Quarter Circle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • circle quarter 4A part of the interior of a circle having two radius boundries at a 90° angle and an arc.
  • Center of a circle having all points on the line circumference are at equal distance from the center point.
  • A quarter circle is a structural shape used in construction.
  • See Geometric Properties of Structural Shapes

area of a Quarter Circle  formula

\(\large{ A =  \frac   { \pi\; r^2 } {4}   }\)

\(\large{ A = \frac { \pi \;d^2} {16} }\)

Where:

\(\large{ A }\) = area

\(\large{ d }\) = diameter

\(\large{ r }\) = radius

\(\large{ \pi }\) = Pi

Circumference of a Quarter Circle formula

\(\large{ C = \frac {\pi \;d} {4}  }\)

Where:

\(\large{ C }\) = circumference

\(\large{ d }\) = diameter

\(\large{ \pi }\) = Pi

Perimeter of a Quarter Circle formula

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

\(\large{ P =   \frac    { \pi \;d }  { 4 }   + d   }\)

Where:

\(\large{ P }\) = perimeter

\(\large{ d }\) = diameter

\(\large{ r }\) = radius

\(\large{ \pi }\) = Pi

Radius of a Quarter Circle formula

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

Where:

\(\large{ r }\) = radius

\(\large{ A }\) = area

\(\large{ \pi }\) = Pi

Distance from Centroid of a Quarter Circle formula

\(\large{ C_x =  \frac {4\;r} {3\; \pi}   }\)

\(\large{ C_y =  \frac   {4\;r}  {3\; \pi}   }\)

Where:

\(\large{ C_x, C_y }\) = distance from centroid

\(\large{ r }\) = radius

\(\large{ \pi }\) = Pi

Elastic Section Modulus of a Quarter Circle formula

\(\large{ S =  \frac { I_x }  { C_y  }  }\)

Where:

\(\large{ S }\) = elastic section modulus

\(\large{ I }\) = moment of inertia

Polar Moment of Inertia of a Circle formula

\(\large{ J_{z} = \left(    \frac { \pi}{ 8 } - \frac { 8 }{ 9\; \pi } \right)   r^4   }\)

\(\large{ J_{z1} =  \frac { \pi \;r^4 }  {  8 } }\)

Where:

\(\large{ J }\) = torsional constant

\(\large{ r }\) = radius

\(\large{ \pi }\) = Pi

Radius of Gyration of a Half Circle formula

\(\large{ k_{x} =    r  \; \sqrt {       \frac { 1 } { 4  }   -  \frac { 16 } { 9\; \pi^2  }      }    }\)

\(\large{ k_{y} =   r  \; \sqrt {       \frac { 1 } { 4  }   -  \frac { 16 } { 9\; \pi^2  }      }    }\)

\(\large{ k_{z} =   r  \; \sqrt {       \frac { 1 } { 2  }   -  \frac { 16 } { 9\; \pi^2  }      }      }\)

\(\large{ k_{x1} =   \frac {  r  }  { 2  }    }\)

\(\large{ k_{y1} =   \frac {  r  }  { 2  }   }\)

\(\large{ k_{z1} =   \frac {  \sqrt {2}  }  { 2  } \; r   }\)

Where:

\(\large{ k }\) = radius of gyration

\(\large{ r }\) = radius

\(\large{ \pi }\) = Pi

Second Moment of Area of a Half circle formula

\(\large{ I_{x} =   \left(    \frac { \pi}{ 16 } - \frac { 4 }{ 9 \;\pi } \right)  \; r^4     }\)

\(\large{ I_{y} =  \left(    \frac { \pi}{ 16 } - \frac { 4 }{ 9\; \pi } \right) \;  r^4 }\)

\(\large{ I_{x1} =   \frac {  \pi\; r^4}{ 16 }  }\)

\(\large{ I_{y1} =  \frac { \pi\; r^4}{ 8 }  }\)

Where:

\(\large{ I }\) = moment of inertia

\(\large{ r }\) = radius

\(\large{ \pi }\) = Pi

 

Tags: Equations for Moment of Inertia Equations for Structural Steel Equations for Modulus