How are these calculations done?

Backing line calculation

The lenght \( L_3 \) of backing line is calculated as follows:

\[ L_3 = \frac{L_1 d_1^2 - L_2 d_2^2}{d_3^2} \]

where \( L_1 \) is the maximum length of diamter \( d_1 \) line your reel can hold, \( L_2 \) and \( d_2 \) is the length and diameter respectively of your main line, and \( d_3 \) is the diameter of your backing line.

For example. Assume that you know that your reel can hold \(L_1=205 \) meters of diameter \( d_1 = 0.29 \) line, and that you want to fill it with \( L_2 = 100 \) meters of diameter \( d_2 = 0.32 \) mm line, plus some unknown length \( L_3 \) of backing line of diameter \( d_3 = 0.40 \) mm. Then

\[ L_3=\frac{205 \times 0.29^2 - 100 \times 0.32^2}{0.40^2} \approx 44 \]

That is, you should use 44 meters of backing line if you want to fill your reel.

If you're interested in knowing how the above formula is derived, consider that the volume \( V_i \) of line \( i \) of length \( L_i \) and diameter \( d_i \), is the length times the cross-sectional area:

\[ V_i = L_i \pi \frac{d_i^2}{4} \]

Keep in mind that this volume is constant if the line is straight or spooled on a reel. Assuming now that you know that your reel holds \( L_1 \) units of diameter \( d_1 \) line and that you want to spool it with \( L_2 \) units of diameter \( d_2 \) main line plus some suitable length \( L_3\) of diameter \( d_3 \) backing line. Then the volume \( V_1 \) of line 1 should equal the sum of volumes \( V_2 \) and \( V_3 \) of the main line 2 and backing line 3. That is

\[ V_1 = V_2 + V_3 \]

Substituting the expression for line volume \( V_i \) above, we get

\[ L_1 \pi \frac{d_1^2}{4} = L_2 \pi \frac{d_2^2}{4} + L_3 \pi \frac{d_3^2}{4} \]

Getting rid of the common factors \( \pi \) and \( 1/4 \) and solving for the length \( L_3 \) of backing line, yields the stated formula

\[ L_3 = \frac{L_1 d_1^2 - L_2 d_2^2}{d_3^2} \]

Line capacity conversion

Assuming that we have zero length backing line, we can solve for \( L_2 \) to get the line capacity conversion formula

\[ L_2 = \frac{L_1 d_1^2}{d_2^2} \]

Hence, if we're interested in 0.32 mm line and know that our reel holds 205 meters of 0.29 mm line, we can use this formula to see that

\[ L_2 = \frac{205 \times 0.29^2}{0.32^2} \approx 168 \]

That is, the reel will hold roughly 168 meters of 0.32 mm line.