Circle Sector

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • circle sector 5Sector of a circle (a two-dimensional figure) is a fraction of the area of a circle with a radius on each side and an arc.
  • Center of a circle having all points on the line circumference are at equal distance from the center point.
  • A half circle is a structural shape used in construction.

Structural Shapes

Arc Length of a Circle Sector formula

\(\large{ l =   \theta \; \frac{\pi}{180} \; r }\)

Where:

\(\large{ l }\) = arc length

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

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

\(\large{ r }\) = radius

area of a Circle Sector formula

\(\large{ A_{area} =  \theta \;r^2  }\)

Where:

\(\large{ A_{area} }\) = area

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

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

\(\large{ r }\) = radius

Perimeter of a Circle Sector formula

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

Where:

\(\large{ P }\) = perimeter

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

\(\large{ r }\) = radius

Distance from Centroid of a Circle Sector formula

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

\(\large{ C_y =  0  }\)

Where:

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

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

\(\large{ r }\) = radius

Elastic Section Modulus of a Circle Sector formula

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

Where:

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

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

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

\(\large{ r }\) = radius

Polar Moment of Inertia of a Circle Sector formula

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

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

Where:

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

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

\(\large{ r }\) = radius

Radius of Gyration of a Circle Sector formula

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

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

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

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

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

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

Where:

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

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

\(\large{ r }\) = radius

Second Moment of Area of a Circle Sector formula

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

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

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

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

Where:

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

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

\(\large{ r }\) = radius

 

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