#"Given:"#
#"Angle"theta_A=pi/6#
#"Angle"theta_B=pi/12#
#"Area of triangle"=15#
#"To find:"#
#"Area of the triangle's incircle"#
#theta_A+theta_B+theta_C=pi#
#theta_A+theta_B=pi/6+pi/12#
#pi/6+pi/12=pi/4#
#theta_A+theta_B=pi/4#
#pi/4+theta_C=pi#
#theta_C=pi-pi/4#
#pi-pi/4=(3pi)/4#
#theta_C=(3pi)/4#
#"Let " r " be the radius of the triangle's incircle"#
#"Area of the triangle's incircle"=pir^2#
#"Vertex Angle at A is "theta_A=pi/6#
#"Semi-Vertex angle at A is "=pi/12#
#"tangent length at vertex A is "l_A=rtan(pi/12)#
#"Vertex Angle at B is "theta_B=pi/12#
#"Semi-Vertex angle at B is "=pi/24#
#"tangent length at vertex B is "l_B=rtan(pi/24)#
#"Vertex Angle at C is "theta_C=(3pi)/4#
#"Semi-Vertex angle at C is "=(3pi)/8#
#"tangent length at vertex C is "l_C=rtan((3pi)/8)#
#"Length of side AB is "=l_A+l_B=rtan(pi/12)+rtan(pi/24)#
#"Length of side AB is "=r(tan(pi/12)+tan(pi/24))#
#"Length of side AC is "=l_A+l_C=rtan(pi/12)+rtan((3pi)/8)#
#"Length of side AC is "=r(tan(pi/12)+tan((3pi)/8))#
#"Vertex Angle at A is "A=pi/6#
#"Area of triangle is "A=15#
#"Area of triangle "=1/2xxABxxACxxsintheta_A#
#"Substituting"#
#15=1/2xx(r(tan(pi/12)+tan(pi/24)))xx(r(tan(pi/12)+tan((3pi)/8)))#
#15=r^2/2xx(tan(pi/12)+tan(pi/24))(tan(pi/12)+tan((3pi)/8))#
#r^2/2xx(tan(pi/12)+tan(pi/24))(tan(pi/12)+tan((3pi)/8))=15#
#r^2/2=15/((tan(pi/12)+tan(pi/24))(tan(pi/12)+tan((3pi)/8)))#
#"Area of the triangle's incircle"=pir^2#
#r^2=30/((tan(pi/12)+tan(pi/24))(tan(pi/12)+tan((3pi)/8)))#
#"Area of the triangle's incircle"=pixx30/((tan(pi/12)+tan(pi/24))(tan(pi/12)+tan((3pi)/8)))#
#"Area of the triangle's incircle"=(30pi)/((tan(pi/12)+tan(pi/24))(tan(pi/12)+tan((3pi)/8)))#
#"Area of the triangle's incircle "=87.934#