Abstract: An improvement of the convergence of triangular quadrature elements is achieved via the increase of quadrature precision. Taking a planar triangular grid points mapped from equi-areal-point-set on a spherical triangle as initial values and the highest order of polynomials that can be accurately integrated as the objective, a new set of quadrature points and weights in a triangle is obtained using numerical optimization methods. The new quadrature rule raises the quadrature precision from the order of the interpolation by approximately 1.6 times. Preliminary numerical examples show that the new triangular quadrature element exhibits a higher convergence rate. Further improvements and possible limitations of this strategy are also discussed.