# Sector of a Circle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

•  Sector is a fraction of the area of a circle with a radius on each side and an arc.
• Angle ($$\Delta$$)  -  Two rays sharing a common point.
• Center (cp)  -  Having all points on the line circumference are at equal distance from the center point.
• Chord (c)  -  Also called long chord (LC), is between any two points on a circular curve.
• Circumference (C)  -  The outside of a circle or a complete circular arc.
• Height (h)  -  Length of radius from radius center to midpoint of chord.
• Height (h')  -  Length of radius from midpoint of chord to point on circular curve.
• Length (L)  -  Total length of any circular curve measured along the arc.
• Radius (r)  -  Half the diameter of a circle.  A line segment between the center point and a point on a circle or sphere.
• Radius Point (rp)  -  Radius center point of circular curve.
• Segment is an interior part of a circle bound by a chord and an arc.
• Tangent (T)  -  A line that touches a curve at just one point such that it is perpendicular to a radius line of the curve.

## Angle of a Sector formula

 $$\large{ \Delta = \frac{ 2 \; A }{r^2} }$$

### Where:

 Units English Metric $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ A }$$ = area of sector $$\large{ in^2 }$$ $$\large{ mm^2 }$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Arc Length of a Sector formula

 $$\large{ L = \Delta \; \frac{\pi}{180} \; r }$$

### Where:

 Units English Metric $$\large{ L }$$ = arc length $$\large{ in }$$ $$\large{ mm }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## area of a Sector formula

 $$\large{ A = \frac{ \Delta \;r^2 }{2} }$$ $$\large{ A = \frac { \Delta } { 360 } \; \pi \; r^2 \;\; }$$ $$\large{ A = \frac { \Delta \; \pi } { 360 } \; r^2 \;\; }$$

### Where:

 Units English Metric $$\large{ A }$$ = area of sector $$\large{ in^2 }$$ $$\large{ mm^2 }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Distance from Centroid of a Sector formulas

 $$\large{ C_x = 2 \; r \; \frac{sin \; \Delta}{3\; \theta} }$$ $$\large{ C_y = 0 }$$

### Where:

 Units English Metric $$\large{ C_x, C_y }$$ = distance from centroid $$\large{ in }$$ $$\large{ mm }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Elastic Section Modulus of a Sector formula

 $$\large{ S = \frac{ I_x }{ sin \; \Delta \; r } }$$

### Where:

 Units English Metric $$\large{ S }$$ = elastic section modulus $$\large{ in^4 }$$ $$\large{ mm^4 }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$ $$\large{ mm^4 }$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Perimeter of a Sector formula

 $$\large{ P = 2 \; r + 2 \; r \; \Delta }$$

### Where:

 Units English Metric $$\large{ P }$$ = perimeter $$\large{ in }$$ $$\large{ mm }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Polar Moment of Inertia of a Sector formulas

 $$\large{ J_{z} = \frac {r^4}{18} \; \left( \frac {9 \; \Delta^2 \;-\; 8 \; sin^2 \; \Delta }{\Delta} \right) }$$ $$\large{ J_{z1} = \frac {r^4 \; \Delta}{2} }$$

### Where:

 Units English Metric $$\large{ J }$$ = torsional constant $$\large{ in^4 }$$ $$\large{ mm^4 }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Radius of a Sector formula

 $$\large{ r = \sqrt{ \frac{ 2 \; A }{\Delta} } }$$

### Where:

 Units English Metric $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$ $$\large{ A }$$ = area of sector $$\large{ in^2 }$$ $$\large{ mm^2 }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$

## Radius of Gyration of a Sector formulas

 $$\large{ k_{x} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \theta \;-\; sin \; \left(2 \; \theta \right) }{\theta} } }$$ $$\large{ k_{y} = \frac{1}{12} \; \sqrt { 2 \; r^2 \; \frac{180^2 \; + \; 9 \; \theta \; sin \; \left(2 \; \theta \right) \;-\; 32 \; + \; 32 \; cos^2 \; \theta }{\theta^2} } }$$ $$\large{ k_{z} = \frac{1}{6} \; \sqrt { 2 \; r^2 \; \frac{9 \; \Delta^2 \;-\; 8 \; sin^2 \; \left(2\; \Delta \right) }{\Delta^2} } }$$ $$\large{ k_{x1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \Delta \;-\; sin \; \left(2 \; \Delta \right) }{\Delta} } }$$ $$\large{ k_{y1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \Delta \; + \; sin \; \left(2 \; \Delta \right) }{\Delta} } }$$ $$\large{ k_{x1} = \frac{r}{ \sqrt{2} } }$$

### Where:

 Units English Metric $$\large{ k }$$ = radius of gyration $$\large{ in }$$ $$\large{ mm }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Second Moment of Area of a Sector formulas

 $$\large{ I_{x} = \frac{r^4}{4} \; \left[ \Delta \;-\; \frac{1}{2} \; sin \left( 2 \; \Delta \right) \right] }$$ $$\large{ I_{y} = \frac{r^4}{4} \; \left[ \Delta + \frac{1}{2} \; sin \left( 2 \; \Delta \right) \right] \;-\; \frac{4r^4}{9 \Delta} \; sin^2 \; \Delta }$$ $$\large{ I_{x1} = I_x + r^4 \; \Delta \; sin^2 \; \Delta }$$ $$\large{ I_{y1} = \frac{r^4}{4} \left[ \Delta + \frac{1}{2} \; sin \; \left( 2 \; \Delta \right) \right] }$$

### Where:

 Units English Metric $$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$ $$\large{ mm^4 }$$ $$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$ $$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$ 