"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
