Question 1133723

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Since {{{AB}}}, {{{AC}}} and {{{BC}}} are all radii, so they are all equal, so {{{ABC}}} is an equilateral triangles and all the angles are {{{60}}} degrees.

 area of triangle {{{ABC=(1/2)ab*sin(60)}}}........{{{a=r}}},{{{ b=r}}}

{{{ABC=(1/2)r^2*(sqrt(3)/2)}}}

{{{ABC=(r^2*sqrt(3))/4}}}


Now I want to know what the area of minor segment{{{ AB}}} is on the circle centered at {{{C}}}:

segment area={{{ (60/360)*r^2*pi -(r^2*sqrt(3))/4}}}

segment area= {{{(2/12)*r^2*pi -(3r^2*sqrt(3))/12}}}

segment area= {{{(2*r^2*pi -3r^2*sqrt(3))/12}}}

=> now

shadedarea={{{3*(2*r^2*pi -3r^2*sqrt(3))/12 +(r^2*sqrt(3))/4}}}

shadedarea={{{cross(3)*(2*r^2*pi -3r^2*sqrt(3))/cross(12)4 +(r^2*sqrt(3))/4}}}

shadedarea={{{(2*r^2*pi -3r^2*sqrt(3))/4 +(r^2*sqrt(3))/4}}}

shadedarea={{{(2*r^2*pi -3r^2*sqrt(3) +r^2*sqrt(3))/4}}}

shadedarea={{{(2*r^2*pi -2r^2*sqrt(3))/4}}}

shadedarea={{{(cross(2)*r^2*pi -cross(2)r^2*sqrt(3))/cross(4)2}}}

shadedarea={{{(r^2*pi -r^2*sqrt(3))/2}}}

shadedarea={{{(r^2/2)(pi -sqrt(3))}}}

 
if {{{r=2cm}}}, the area of the shaded region is:

{{{shadedarea=((2cm)^2/2)(pi-sqrt(3))}}}

{{{shadedarea=(4cm^2/2)(pi-sqrt(3))}}}

{{{shadedarea= 2cm^2(pi-sqrt(3))}}}

{{{shadedarea=2.8190cm^2}}}