Question 1206037
<pre>

I'm going to guess what your problem is.  I'll bet there are two concentric
circles, the smaller circle has a radius of OP = 6 cm. Chord AB is tangent to
the smaller circle at P. And AB = {{{12sqrt(3)}}}cm.  Find the area of the 
shaded region, the shaded segment of the larger circle.  Isn't that what
you want?

{{{drawing(400,400,-13,13,-13,13,
locate(6.02,11.3,A),locate(6.02,-10.6,B),locate(-.2,-.2,O),locate(5.4,-.2,P),
circle(0,0,6), circle(0,0,12),line(-12,0,6,0),
circle(0,0,.25),

arc(0,0,24.3,-24.3,300,420),arc(0,0,24.3,-24.3,60,300),

line(6,-6sqrt(3),6,6sqrt(3)),
green(

line(6.21,-10.2681985,6.21,10.2681985),line(6.28,-10.2255367,6.28,10.2255367),line(6.35,-10.1822149,6.35,10.1822149),line(6.35,-10.1822149,6.35,10.1822149),
line(6.42,-10.1382247,6.42,10.1382247),line(6.49,-10.0935574,6.49,10.0935574),line(6.56,-10.0482038,6.56,10.0482038),line(6.56,-10.0482038,6.56,10.0482038),
line(6.63,-10.0021548,6.63,10.0021548),line(6.7,-9.95540054,6.7,9.95540054),line(6.77,-9.90793117,6.77,9.90793117),line(6.77,-9.90793117,6.77,9.90793117),
line(6.84,-9.8597363,6.84,9.8597363),line(6.91,-9.81080527,6.91,9.81080527),line(6.98,-9.76112698,6.98,9.76112698),line(6.98,-9.76112698,6.98,9.76112698),
line(7.05,-9.71068999,7.05,9.71068999),line(7.12,-9.65948239,7.12,9.65948239),line(7.19,-9.60749187,7.19,9.60749187),line(7.19,-9.60749187,7.19,9.60749187),
line(7.26,-9.55470565,7.26,9.55470565),line(7.33,-9.50111046,7.33,9.50111046),line(7.4,-9.44669254,7.4,9.44669254),line(7.4,-9.44669254,7.4,9.44669254),
line(7.47,-9.39143759,7.47,9.39143759),line(7.54,-9.33533074,7.54,9.33533074),line(7.61,-9.27835654,7.61,9.27835654),line(7.61,-9.27835654,7.61,9.27835654),
line(7.68,-9.2204989,7.68,9.2204989),line(7.75,-9.1617411,7.75,9.1617411),line(7.82,-9.1020657,7.82,9.1020657),line(7.82,-9.1020657,7.82,9.1020657),
line(7.89,-9.04145453,7.89,9.04145453),line(7.96,-8.97988864,7.96,8.97988864),line(8.03,-8.91734826,8.03,8.91734826),line(8.03,-8.91734826,8.03,8.91734826),
line(8.1,-8.85381274,8.1,8.85381274),line(8.17,-8.78926049,8.17,8.78926049),line(8.24,-8.72366895,8.24,8.72366895),line(8.24,-8.72366895,8.24,8.72366895),
line(8.31,-8.6570145,8.31,8.6570145),line(8.38,-8.58927238,8.38,8.58927238),line(8.45,-8.52041666,8.45,8.52041666),line(8.45,-8.52041666,8.45,8.52041666),
line(8.52,-8.45042011,8.52,8.45042011),line(8.59,-8.37925414,8.59,8.37925414),line(8.66,-8.30688871,8.66,8.30688871),line(8.66,-8.30688871,8.66,8.30688871),
line(8.73,-8.23329217,8.73,8.23329217),line(8.8,-8.15843122,8.8,8.15843122),line(8.87,-8.08227072,8.87,8.08227072),line(8.87,-8.08227072,8.87,8.08227072),
line(8.94,-8.00477358,8.94,8.00477358),line(9.01,-7.92590058,9.01,7.92590058),line(9.08,-7.84561024,9.08,7.84561024),line(9.08,-7.84561024,9.08,7.84561024),
line(9.15,-7.76385858,9.15,7.76385858),line(9.22,-7.68059893,9.22,7.68059893),line(9.29,-7.59578172,9.29,7.59578172),line(9.29,-7.59578172,9.29,7.59578172),
line(9.36,-7.50935417,9.36,7.50935417),line(9.43,-7.42126,9.43,7.42126),line(9.5,-7.33143915,9.5,7.33143915),line(9.5,-7.33143915,9.5,7.33143915),
line(9.57,-7.23982735,9.57,7.23982735),line(9.64,-7.14635571,9.64,7.14635571),line(9.71,-7.05095029,9.71,7.05095029),line(9.71,-7.05095029,9.71,7.05095029),
line(9.78,-6.95353148,9.78,6.95353148),line(9.85,-6.85401342,9.85,6.85401342),line(9.92,-6.75230331,9.92,6.75230331),line(9.92,-6.75230331,9.92,6.75230331),
line(9.99,-6.64830053,9.99,6.64830053),line(10.06,-6.54189575,10.06,6.54189575),line(10.13,-6.43296977,10.13,6.43296977),line(10.13,-6.43296977,10.13,6.43296977),
line(10.2,-6.32139225,10.2,6.32139225),line(10.27,-6.20702022,10.27,6.20702022),line(10.34,-6.08969622,10.34,6.08969622),line(10.34,-6.08969622,10.34,6.08969622),
line(10.41,-5.96924618,10.41,5.96924618),line(10.48,-5.84547688,10.48,5.84547688),line(10.55,-5.71817279,10.55,5.71817279),line(10.55,-5.71817279,10.55,5.71817279),
line(10.62,-5.58709227,10.62,5.58709227),line(10.69,-5.45196295,10.69,5.45196295),line(10.76,-5.31247588,10.76,5.31247588),line(10.76,-5.31247588,10.76,5.31247588),
line(10.83,-5.16827824,10.83,5.16827824),line(10.9,-5.01896404,10.9,5.01896404),line(10.97,-4.86406209,10.97,4.86406209),line(10.97,-4.86406209,10.97,4.86406209),
line(11.04,-4.70302031,11.04,4.70302031),line(11.11,-4.53518467,11.11,4.53518467),line(11.18,-4.35977064,11.18,4.35977064),line(11.18,-4.35977064,11.18,4.35977064),
line(11.25,-4.17582327,11.25,4.17582327),line(11.32,-3.98216022,11.32,3.98216022),line(11.39,-3.77728739,11.39,3.77728739),line(11.39,-3.77728739,11.39,3.77728739),
line(11.46,-3.55926959,11.46,3.55926959),line(11.53,-3.32552252,11.53,3.32552252),line(11.6,-3.0724583,11.6,3.0724583),line(11.6,-3.0724583,11.6,3.0724583),
line(11.67,-2.79483452,11.67,2.79483452),line(11.74,-2.48443152,11.74,2.48443152),line(11.81,-2.12694617,11.81,2.12694617),line(11.81,-2.12694617,11.81,2.12694617),
line(11.88,-1.69280832,11.88,1.69280832),line(11.95,-1.09430343,11.95,1.09430343))

)}}}

Draw the two radii OA and OB of the larger circle (in red):

{{{drawing(400,400,-13,13,-13,13,
red(line(0,0,6,6sqrt(3)),line(0,0,6,-6sqrt(3))),
locate(6.02,11.3,A),locate(6.02,-10.6,B),locate(-.2,-.2,O),locate(5.4,-.2,P),
circle(0,0,6), circle(0,0,12),line(-12,0,6,0),
circle(0,0,.25),
line(6,-6sqrt(3),6,6sqrt(3)),green(

line(6.21,-10.2681985,6.21,10.2681985),line(6.28,-10.2255367,6.28,10.2255367),line(6.35,-10.1822149,6.35,10.1822149),line(6.35,-10.1822149,6.35,10.1822149),
line(6.42,-10.1382247,6.42,10.1382247),line(6.49,-10.0935574,6.49,10.0935574),line(6.56,-10.0482038,6.56,10.0482038),line(6.56,-10.0482038,6.56,10.0482038),
line(6.63,-10.0021548,6.63,10.0021548),line(6.7,-9.95540054,6.7,9.95540054),line(6.77,-9.90793117,6.77,9.90793117),line(6.77,-9.90793117,6.77,9.90793117),
line(6.84,-9.8597363,6.84,9.8597363),line(6.91,-9.81080527,6.91,9.81080527),line(6.98,-9.76112698,6.98,9.76112698),line(6.98,-9.76112698,6.98,9.76112698),
line(7.05,-9.71068999,7.05,9.71068999),line(7.12,-9.65948239,7.12,9.65948239),line(7.19,-9.60749187,7.19,9.60749187),line(7.19,-9.60749187,7.19,9.60749187),
line(7.26,-9.55470565,7.26,9.55470565),line(7.33,-9.50111046,7.33,9.50111046),line(7.4,-9.44669254,7.4,9.44669254),line(7.4,-9.44669254,7.4,9.44669254),
line(7.47,-9.39143759,7.47,9.39143759),line(7.54,-9.33533074,7.54,9.33533074),line(7.61,-9.27835654,7.61,9.27835654),line(7.61,-9.27835654,7.61,9.27835654),
line(7.68,-9.2204989,7.68,9.2204989),line(7.75,-9.1617411,7.75,9.1617411),line(7.82,-9.1020657,7.82,9.1020657),line(7.82,-9.1020657,7.82,9.1020657),
line(7.89,-9.04145453,7.89,9.04145453),line(7.96,-8.97988864,7.96,8.97988864),line(8.03,-8.91734826,8.03,8.91734826),line(8.03,-8.91734826,8.03,8.91734826),
line(8.1,-8.85381274,8.1,8.85381274),line(8.17,-8.78926049,8.17,8.78926049),line(8.24,-8.72366895,8.24,8.72366895),line(8.24,-8.72366895,8.24,8.72366895),
line(8.31,-8.6570145,8.31,8.6570145),line(8.38,-8.58927238,8.38,8.58927238),line(8.45,-8.52041666,8.45,8.52041666),line(8.45,-8.52041666,8.45,8.52041666),
line(8.52,-8.45042011,8.52,8.45042011),line(8.59,-8.37925414,8.59,8.37925414),line(8.66,-8.30688871,8.66,8.30688871),line(8.66,-8.30688871,8.66,8.30688871),
line(8.73,-8.23329217,8.73,8.23329217),line(8.8,-8.15843122,8.8,8.15843122),line(8.87,-8.08227072,8.87,8.08227072),line(8.87,-8.08227072,8.87,8.08227072),
line(8.94,-8.00477358,8.94,8.00477358),line(9.01,-7.92590058,9.01,7.92590058),line(9.08,-7.84561024,9.08,7.84561024),line(9.08,-7.84561024,9.08,7.84561024),
line(9.15,-7.76385858,9.15,7.76385858),line(9.22,-7.68059893,9.22,7.68059893),line(9.29,-7.59578172,9.29,7.59578172),line(9.29,-7.59578172,9.29,7.59578172),
line(9.36,-7.50935417,9.36,7.50935417),line(9.43,-7.42126,9.43,7.42126),line(9.5,-7.33143915,9.5,7.33143915),line(9.5,-7.33143915,9.5,7.33143915),
line(9.57,-7.23982735,9.57,7.23982735),line(9.64,-7.14635571,9.64,7.14635571),line(9.71,-7.05095029,9.71,7.05095029),line(9.71,-7.05095029,9.71,7.05095029),
line(9.78,-6.95353148,9.78,6.95353148),line(9.85,-6.85401342,9.85,6.85401342),line(9.92,-6.75230331,9.92,6.75230331),line(9.92,-6.75230331,9.92,6.75230331),
line(9.99,-6.64830053,9.99,6.64830053),line(10.06,-6.54189575,10.06,6.54189575),line(10.13,-6.43296977,10.13,6.43296977),line(10.13,-6.43296977,10.13,6.43296977),
line(10.2,-6.32139225,10.2,6.32139225),line(10.27,-6.20702022,10.27,6.20702022),line(10.34,-6.08969622,10.34,6.08969622),line(10.34,-6.08969622,10.34,6.08969622),
line(10.41,-5.96924618,10.41,5.96924618),line(10.48,-5.84547688,10.48,5.84547688),line(10.55,-5.71817279,10.55,5.71817279),line(10.55,-5.71817279,10.55,5.71817279),
line(10.62,-5.58709227,10.62,5.58709227),line(10.69,-5.45196295,10.69,5.45196295),line(10.76,-5.31247588,10.76,5.31247588),line(10.76,-5.31247588,10.76,5.31247588),
line(10.83,-5.16827824,10.83,5.16827824),line(10.9,-5.01896404,10.9,5.01896404),line(10.97,-4.86406209,10.97,4.86406209),line(10.97,-4.86406209,10.97,4.86406209),
line(11.04,-4.70302031,11.04,4.70302031),line(11.11,-4.53518467,11.11,4.53518467),line(11.18,-4.35977064,11.18,4.35977064),line(11.18,-4.35977064,11.18,4.35977064),
line(11.25,-4.17582327,11.25,4.17582327),line(11.32,-3.98216022,11.32,3.98216022),line(11.39,-3.77728739,11.39,3.77728739),line(11.39,-3.77728739,11.39,3.77728739),
line(11.46,-3.55926959,11.46,3.55926959),line(11.53,-3.32552252,11.53,3.32552252),line(11.6,-3.0724583,11.6,3.0724583),line(11.6,-3.0724583,11.6,3.0724583),
line(11.67,-2.79483452,11.67,2.79483452),line(11.74,-2.48443152,11.74,2.48443152),line(11.81,-2.12694617,11.81,2.12694617),line(11.81,-2.12694617,11.81,2.12694617),
line(11.88,-1.69280832,11.88,1.69280832),line(11.95,-1.09430343,11.95,1.09430343))

)}}}

{{{PA=expr(1/2)AB=expr(1/2)*12sqrt(3)=6sqrt(3)}}}

Use the Pythagorean theorem on right triangle OPA.
{{{OA^2=OP^2+PA^2}}}
{{{OA^2=6^2+(6sqrt(3))^2}}}
{{{OA^2=36+36*3}}}
{{{OA^2=36+108}}}
{{{OA^2=144}}}
{{{OA=sqrt(144)}}}
{{{OA=12}}} = radius of larger circle.

Triangle OPA is a 30-60-90 right triangle because its shortest side OP=6 cm
and its longest side OA, is 12, twice the shortest side.

So angle AOP = 60<sup>o</sup> = &pi;/3 radians, and angle AOB = 120<sup>o</sup> = 2&pi;/3 radians.
We find the area of the sector OAB and subtract the area of triangle OAB.

Area of the sector is {{{A=expr(1/2)*theta*radius^2=expr(1/2)*expr((2pi)/3)*12^2= expr(1/2)*expr((2pi)/3)*144=48pi}}}

Area of triangle AOP = {{{expr(1/2)*AP*PB=expr(1/2)*6*6*sqrt(3)=18sqrt(3)}}}

Area of triangle AOB = twice area of AOP = {{{36sqrt(3)}}}

Subtract the area of the triangle from the area of the sector:

{{{48pi-36sqrt(3)}}}{{{""=""}}}{{{12(4pi-3sqrt(3))}}}

About 88.4 cm<sup>2</sup>.

Edwin</pre>