# District sector

A circular sector (also circular cutout ) is in the geometry of the partial surface of a circular area , of a circular arc and two circle radii is limited (as opposed to by an arc and a chord "limited circle segment / circuit section "). Viewed from above, this area looks like a piece of cake .

Formulas for the circular sector in degrees
Length of the associated circular arc ${\ displaystyle L = 2r \ cdot \ pi \ cdot {\ frac {\ theta} {360 ^ {\ circ}}}}$
Area ${\ displaystyle A = r ^ {2} \ cdot \ pi \ cdot {\ frac {\ theta} {360 ^ {\ circ}}} = {\ frac {1} {2}} L \ cdot r}$
radius ${\ displaystyle r}$
Center angle (also central angle) ( degree ) ${\ displaystyle \ theta}$
Circle number ${\ displaystyle \ pi \ approx 3 {,} 1415926536}$
Formulas for the circular sector in radians
Length of the associated circular arc ${\ displaystyle L = r \ cdot \ theta}$
Area ${\ displaystyle A = {\ frac {1} {2}} {L \ cdot r} = {\ frac {1} {2}} {r ^ {2}} \ cdot \ theta}$
radius ${\ displaystyle r}$
Center angle ( radians ) ${\ displaystyle \ theta}$
Chord length between the extreme points ${\ displaystyle C = 2r \ cdot \ sin {\ frac {\ theta} {2}}}$

## Area

The area of ​​a circle sector can be derived from the following integral:

${\ displaystyle A = \ int _ {0} ^ {\ theta} \ int _ {0} ^ {r} dS = \ int _ {0} ^ {\ theta} \ int _ {0} ^ {r} { \ tilde {r}} \, d {\ tilde {r}} \, d {\ tilde {\ theta}} = \ int _ {0} ^ {\ theta} {\ frac {1} {2}} r ^ {2} \, d {\ tilde {\ theta}} = {\ frac {r ^ {2} \ theta} {2}}}$