# Circle Segment

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Circle segment is an interior part of a circle bound by a chord and an arc.
• Center of a circle having all points on the line circumference are at equal distance from the center point.

## Formulas that use Arc Length of a Circle Segment

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

### Where:

$$\large{ l }$$ = arc length

$$\large{ \theta }$$ = angle

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

## Formulas that use area of a Circle Segment

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

### Where:

$$\large{ A_{area} }$$ = area

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius

## Formulas that use Distance from Centroid of a Circle Segment

 $$\large{ C_x = 0 }$$ $$\large{ C_y = \frac {4 \; r}{3} \; \left( \frac {sin^3 \; \frac{\theta}{2} } {\theta \; - \; sin \; \theta} \right) }$$

### Where:

$$\large{ C_x, C_y }$$ = distance from centroid

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius

## Formulas that use Elastic Section Modulus of a Circle Segment

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

### Where:

$$\large{ S }$$ = elastic section modulus

$$\large{ C_x, C_y }$$ = distance from centroid

$$\large{ I }$$ = moment of inertia

$$\large{ r }$$ = radius

$$\large{ \theta }$$ = angle

## Formulas that use Perimeter of a Circle Segment

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

### Where:

$$\large{ P }$$ = perimeter

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius

## Formulas that use Polar Moment of Inertia of a Circle Segment

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

### Where:

$$\large{ J }$$ = torsional constant

$$\large{ r }$$ = radius

$$\large{ \theta }$$ = angle

## Formulas that use Radius of Gyration of a Circle Segment

 $$\large{ k_{x} = \sqrt { \frac {I_x}{A_{area}} } }$$ $$\large{ k_{y} = \sqrt { \frac {I_y}{A_{area}} } }$$ $$\large{ k_{z} = \sqrt { k_{x}{^2} + k_{y}{^2} } }$$

### Where:

$$\large{ k }$$ = radius of gyration

$$\large{ A_{area} }$$ = area

$$\large{ I }$$ = moment of inertia

## Formulas that use Second Moment of Area of a Circle Segment

 $$\large{ I_{x} = \frac {r^4}{8} \; \left( \theta - sin \; \theta + 2 \; sin \; \theta \; sin^2 \; \frac {\theta}{2} \right) }$$ $$\large{ I_{y} = \frac {r^4}{24} \; \left( 3 \; \theta - 3 \; sin \; \theta - 2 \; sin \; \theta \; sin^2 \; \frac {\theta}{2} \right) }$$

### Where:

$$\large{ I }$$ = moment of inertia

$$\large{ r }$$ = radius

$$\large{ \theta }$$ = angle