Explanation:
Of all the variables in an amortization formula, the interest rate is the one that cannot be solved for. Finding APR for a given loan is generally an iterative process, or one that can be done by many spreadsheets, graphing calculators, and on-line tools.
The formula for the value of a monthly loan payment A on principal P, for t years, at interest rate r is ...
 A = P(r/12)/(1 -(1 +r/12)^(-12t))
but that is of no direct help in finding APR.
__
This formula can be used to develop relations that can approximate APR. The formulas below are for different loan lengths, and will be accurate to 2 decimal places of APR value for APRs in the range of 0 to 15%.
They work like this:
 Find the ratio of total loan repayment cost to the principal amount of the loan. (Division is required for this.) It will generally be a number between 1 and about 1.5 for auto loans of reasonable duration.
Use that value for x in the expression corresponding to the loan length. The range of output from these formulas is from 0 to 15, meaning the APR is 0% to 15%.
APR Formulas
2-year loan
 APR = 8.03501x - 53.6947)x + 153.636)x - 248.528)x + 306.753)x -166.201
3-year loan
 APR = 4.37626x - 30.7798)x + 92.6767)x - 157.555)x + 203.183)x -111.901
4-year loan
 APR = 2.7783x - 20.3207)x + 63.6117)x - 112.328)x + 150.19)x -83.9314
5-year loan (note the leading minus sign)
 APR = -1.28076x + 11.1732)x - 42.3671)x + 91.6612)x - 125.977)x + 137.601)x -70.8105
6-year loan (note the leading minus sign)
 APR = -0.864078x + 7.82602)x - 30.8075)x + 69.1893)x - 98.6637)x + 111.919)x -58.599
Example
You want to borrow $22,175, and you're told that your monthly payment amount on a 4-year loan would be $512.99. Your total repayment is ...
 $512.99×48 = $24,623.52
Dividing this by the amount you borrowed, you get a ratio of ...
 24,623.52/22,175 ≈ 1.1104
Using this value in the formula for the 4-year loan APR, we make the following calculations:
  APR = 2.7783(1.1104) - 20.3207)(1.1104) + 63.6117)(1.1104) - 112.328)(1.1104) + 150.19)(1.1104) -83.9314
   = (((-17.2357)(1.1104) + 63.6117)(1.1104) - 112.328)(1.1104) + 150.19)(1.1104) -83.9314
   = (((44.4732)(1.1104) - 112.328)(1.1104) + 150.19)(1.1104) -83.9314
   = ((-62.945)(1.1104) + 150.19)(1.1104) -83.9314
   = (80.2959)(1.1104) -83.9314
   = 5.2292
 APR = 5.23 . . . percent (result should be rounded to 2 decimal places)
__
Comment on formula evaluation
It is tedious, but necessary, to keep 4 decimal places of precision in each of the intermediate calculations, rounding where required. These formulas are taking relatively small differences of relatively large numbers, so the best reasonable precision should be maintained in all calculations. The final answer should be rounded to 2 decimal places.
These computations can be done by hand, but a calculator or spreadsheet is immensely helpful.
