I've attached a screen-shot of the spreadsheet:
R1 := radius of the outer circle
R2 := radius of the inner circle
d := length of the secant (with your example 30)
The formula is shown in the input line. Translate the word wurzel by sqrt (Wurzel is the German word for root)
I hope that this helps!
according to the attached pic:
x + y = 30
y = 30 - x .....(1)
By Pythagarus Theorem.:
33^2 - (x + y)^2 = z^2
30^2 - y^2 = z^2
Hence
33^2 - (x + y)^2 = 30^2 - y^2
33^2 - 30^2 = (x + y)^2 - y^2
189 = x^2 + 2xy + y^2 - y^2
x^2 + 2xy = 189 .....(2)
(1) to (2), hence:
x^2 + 2x(30 - x) = 189
x^2 + 60x - 2x^2 - 189 = 0
-x^2 + 60x - 189 = 0
x^2 - 60x + 189 = 0
x = [-(-60) +/- (60^2 - 4 x 1 x 189)^0.5] / (2 x 1)
x = 56.66 (rejected)
or
x = 3.34
alright i am now aware of the problem. however x is not the difference of the two green lines either, because the question has never given that x + y is precisely equivalent to the length of the inner radius (30).
To ignore this arguement, we only use the Pyth. Th. twice:
(x + y)^2 + 15^2 = 33^2
(x + y)^2 = 33^2 - 15^2
x^2 + 2xy + y^2 = 864 .....(1)
y^2 + 15^2 = 30^2
y^2 = 675
y = (5)(27^0.5) .....(2)
(2) to (1) we have:
x^2 + 2x(5)(27^0.5) + 675 = 864
x^2 + (10)(27^0.5)x - 189 = 0
x = [-(10)(27^0.5) +/- (2700 + 4 x 189)^0.5] / 2
x = 3.41
or
x = -55.4 (rejected)