What is the formula for calculating the diameter of the pulleys to change the pitch angle of the stepper?

Question

Geometry

The essence of the problem I want to solve:

• I need to change the angle of rotation on a stepper motor, using a belt drive
  

Who knows the formula by which you can calculate the diameter of the driven pulley knowing the angle by which the drive pulley will turn, the diameter of the pulley, the angle by which the driven pulley should turn?
the only thing that comes to mind is the ratio of the diameter of the drive pulley and the driven one. Let's say that the drive pulley has turned 5.625° in one step, and its diameter is 30mm, and we need the driven pulley to turn 3°.
If we think about it, then we already need to do something else here, calculate the diameter based on the circle.
If I'm right, then if we assume that the drive pulley has a diameter of d1 = 30mm, then its circumference isp=π*d= 2π*r
p1=π*d1= 2π*r1

p1=π*30= 2π*15 = 3.14159 * 30 = 2 * 3.14159 * 15 = 94,28571428571429

now let's calculate the belt travel along the pulley in one step of the motor
, 360° = 94,28571428571429respectively, the belt travel in one step equal to 5.625 ° will be

94,28571428571429 / (360° / 5.625°) = 1,473214285714286

now having a belt travel of one step, let's substitute here our unknown circle of the driven pulley
p2 / (360° / 3°) = 1,473214285714286
, transfer the unknowns to the left side:
p2 = (360° / 3°) * 1,473214285714286
and now we calculate
p2 = 176,7857142857143
based on the formula p = π * d we find the diameter d2 = 176,7857142857143 / π = 56,25000000000001
of the driven pulley, and according to my calculations, with a diameter of the drive pulley of 30 mm, I need the diameter of the driven pulley is 56.25 mm,
based on the calculations and the train of thought, for convenience, let's try to derive a formula for calculating such a gearbox of the angle of rotation of a stepper motor.
Diameter_2 = Diameter_1 * (step_angle_1 / step_angle_2)
hmm I wonder if I calculated correctly, who can tell?

Answer 1

Diameter_2 = Diameter_1 * (pitch_1 angle / pitch_2 angle)