Circle - area of an arc sector

vCalc Reviewed

{Area}_{sector} = (\frac{θ}{2}) \cdot r^{2}

$"Area"_"sector" = (( theta )/2) * "r" ^2$

(r) Radius of Circle

$(r) "Radius of Circle"$

(

$($ theta

) Angle Defining Sector

$) "Angle Defining Sector"$

INPUTS

The inputs defining the circle are:

r - the radius of the circle
$θ$ $theta$ - the central angle of the sector

NOTES

The internal formula computes degrees or radians as:

For Radians: ( $\frac{θ}{2}$ $theta /2$ ) * $\cdot r^{2}$ $*r^2$

For Degrees: ( $\frac{θ}{360}$ $theta / 360$ ) * $π \cdot r^{2}$ $pi * r^2$

The area of a sector of a circle is shown in red in the image at the right. The sector is fully defined by the radius, r, and the central angle,

θ

$theta$ , of the sector.

/attachments/dec4df82-9df3-11e3-9cd9-bc764e2038f2/Arc sector - arc segment.png

The area of a circle is defined as $π$ $pi$ * $r^{2}$ $r^2$ . The area of a full circle can be described as a sector whose central angle encompasses 2 $π$ $pi$ radians (360 degrees) of angular dimension around a full circle. The sector shown in the right image then is a portion of the whole circle and the area is that portion of the whole circle defined by the ratio of the angle $θ$ $theta$ to the angle for the whole circle; i.e., 2 $π$ $pi$ radians (360 degrees).

Thus, the proportion of the circle's area that is the sector's area is the area of the circle multiplied by the ratio of the angular dimension of the sector to the angular dimension of the whole circle:

${Area}_{Sector}$ $"Area"_"Sector"$	$(θ$ $(theta$ radians / $2 π radians) \cdot π \cdot r^{2}$ $2pi " radians") * pi * r^2$ = ( $θ$ $theta$ / 2) * $r^{2}$ $r^2$
	$(θ$ $(theta$ degrees / $360 degrees) \cdot π \cdot r^{2}$ $360 " degrees") * pi * r^2$ = ( $θ$ $theta$ / 360) * $π \cdot r^{2}$ $pi *r^2$

The angular units cancel out leaving in the first multiplicative term either the number $θ$ $theta$ / 2 $π$ $pi$ or $θ$ $theta$ / 360 , corresponding to the angular unit used. That unit-less ratio is then multiplied by the area of the whole circle: the famed $π \cdot r^{2}$ $pi *r^2$ , where r has units of length.