import math

def manning_opt_1(alpha,D):
    A = 1/8.0*(math.sin(alpha) * D**2)
    P = .5*alpha*D
    return {'A':A,'P':P}

print manning_opt_1(240*math.pi/180,4)
#{'A': -1.7320508075688767, 'P': 8.377580409572781}

def manning_opt_2(d,D):
    A = D**2/4*math.acos((D/2.0 -d)/(D/2.0)) - (D/2.0 -d)*math.sqrt(D*d - d**2)
    P = D/2.0*(2*math.pi - 2*math.acos((d-D/2.0)/(D/2.0)))
    return {'A':A,'P':P}

print manning_opt_2(3,4)
#{'A': 10.10963121714166, 'P': 8.377580409572781}