measuring the size and mass of other planets

For any planet with a moon, the planet's mass can be approximated by GmM/r² = mv²/r = mrω² = mr(2π/P)², where G is the gravitational constant, m the mass of the moon, M the mass of the planet, r the distance between planet and moon, and P the period of the moon. The mass of the moon cancels out, leaving
M = r³(2π/P)²/G

The masses of moonless planets have to be determined by Perturbation Theory, the effect of one planet's orbit on the orbits of other planets.

The diameter of another celestial object can be estimated by multiplying the angle subtended by the object by its distance from us. A closer estimate is obtained by
d/2 = Dsin(α/2)
where d is the diameter, D is distance to the object, and α is the subtended angle.