Circle Sector

Written by Jerry Ratzlaff on . Posted in Plane Geometry

Circle Sector - Geometric Propertiescircle sector 4

area of a Circle Sector formula

\(\large{ A =   \theta r^2   }\)

Center of a Circle Sector

All points on the line circumference are at equal distance from the center point.

Perimeter of a Circle Sector formula

\(\large{ P =   2r  +  2r \theta   }\)

Distance from Centroid of a Circle Sector formula

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

\(\large{ C_y =  0   }\)

Elastic Section Modulus of a Circle Sector formula

\(\large{ S =  \frac{ I_x }{ sin \; \left( \theta \right) r  }  }\)

Polar Moment of Inertia of a Circle Sector formula

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

\(\large{ J_{z1} =   \frac {r^4 \theta}{2}     }\)

Radius of Gyration of a Circle Sector formula

\(\large{ k_{x} =   \frac{1}{4}   \sqrt {  2r^2  \frac{2 \theta \;-\; sin \; \left(2 \theta \right) }{\theta}     }   }\)

\(\large{ k_{y} =  \frac{1}{12}   \sqrt {  2r^2  \frac{180^2 \; + \; 9 \theta \; sin \; \left(2 \theta \right) \;-\; 32 \; + \; 32 \; cos^2 \; \left( \theta \right)   }{\theta^2}     }       }\)

\(\large{ k_{z} =  \frac{1}{6}   \sqrt {  2r^2  \frac{9 \theta^2 \;-\; 8 \; sin^2 \; \left(2 \theta \right) }{\theta^2}     }       }\)

\(\large{ k_{x1} =  \frac{1}{4}   \sqrt {  2r^2  \frac{2 \theta \;-\; sin  \; \left(2 \theta \right) }{\theta}     }        }\)

\(\large{ k_{y1} =  \frac{1}{4}   \sqrt {  2r^2  \frac{2 \theta \; + \; sin \; \left(2 \theta \right) }{\theta}     }        }\)

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

Second Moment of Area of a Circle Sector formula

\(\large{ I_{x} =  \frac{r^4}{4}  \left[ \theta  \;-\;  \frac{1}{2} sin \left( 2 \theta \right)    \right]   }\)

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

\(\large{ I_{x1} =  I_x  +  r^4 \theta \; sin^2  \left( \theta \right)     }\)

\(\large{ I_{y1} =  \frac{r^4}{4}  \left[ \theta  +  \frac{1}{2} sin \left( 2 \theta \right)    \right]    }\)

 

Where:

\(\large{ A }\) = area

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

\(\large{ d }\) = diameter

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

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

\(\large{ P }\) = perimeter

\(\large{ r }\) = radius

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

\(\large{ \theta }\) = angle

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

 

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