# Circle Segment

Written by Jerry Ratzlaff on . Posted in Plane Geometry

## Circle Segment - Geometric Properties

### area of a Circle Segment  formula

$$\large{ A = \frac {r^2} {2} \left( \theta \; - \; sin \; \theta \right) }$$

### Center of a Circle Segment

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

### Perimeter of a Circle Segment formula

$$\large{ P = \frac {\pi \; \theta \; r}{180} + 2 r \; sin \; \frac { \theta }{2} }$$

### Distance from Centroid of a Circle Segment formula

$$\large{ C_x = 0 }$$

$$\large{ C_y = \frac {4r}{3} \left( \frac {sin^3 \frac{\theta}{2} } {\theta \; - \; sin \; \theta} \right) }$$

### Elastic Section Modulus of a Circle Segment formula

$$\large{ S = \frac { I_x } { C_y \;-\; r \; cos \; \left( \frac {\theta}{2} \right) } }$$

### Polar Moment of Inertia of a Circle Segment formula

$$\large{ J_{z} = \frac {r^4}{4} \left( \theta \;-\; sin \; \; \theta \; + \; \frac {2}{3} \; sin \; \theta \; sin^2 \frac {\theta}{2} \right) }$$

### Radius of Gyration of a Circle Segment formula

$$\large{ k_{x} = \sqrt { \frac {I_x}{A} } }$$

$$\large{ k_{y} = \sqrt { \frac {I_y}{A} } }$$

$$\large{ k_{z} = \sqrt { k_{x}{^2} + k_{y}{^2} } }$$

### Second Moment of Area of a Circle Segment formula

$$\large{ I_{x} = \frac {r^4}{8} \left( \theta \;-\; sin \; \; \theta \; + \; 2 \; sin \; \theta \; sin^2 \frac {\theta}{2} \right) }$$

$$\large{ I_{y} = \frac {r^4}{24} \left( 3 \; \theta \;-\; 3 \; sin \; \; \theta \; - \; 2 \; sin \; \theta \; sin^2 \frac {\theta}{2} \right) }$$

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