# How does the FAA compute the values in TPP's Climb/Descent Table?

The inside back cover of the FAA Terminal Procedures Publication contains a table headed “CLIMB/DESCENT TABLE,” excerpted below. Source: FAA, p. 19

How do required ascent or descent per nautical mile and ground speed combine to give feet per minute? What is the relationship between the values in the two leftmost columns?

Given a required climb or descent in feet per minute and ground speed in knots, compute the target VSI reading with required climb/descent multiplied by ground speed divided by sixty, or

$$\delta_{VSI} = \frac{\mathrm{RCD} \times \mathrm{GS}}{60}$$

Apply dimensional analysis to see why this works. Informally, we can think of nautical miles and hours in the numerator and denominator as canceling each other out to leave a figure in feet per minute.

$$\delta_{VSI}\ \frac{\mathrm{ft}}{\mathrm{min}} = \mathrm{RCD}\ \frac{\mathrm{ft}}{\mathrm{nm}} \times \mathrm{GS}\ \frac{\mathrm{nm}}{\mathrm{hour}} \times \frac{\mathrm{1\ hour}}{\mathrm{60\ min}}$$

Fundamentally, trigonometry determines the values given angles in the leftmost column and the FAA’s definition of a nautical mile being 6,076.1 feet. In a right triangle, tangent is the ratio of the opposite leg to the adjacent leg for a given angle $\theta$, illustrated below for a climb. For the values in the table

• the adjacent leg is exactly 1 nautical mile
• the angle $\theta$ is the precise value from the leftmost column
• the opposite leg is unknown and is the desired ft/NM value

The core formula is $6,076.1 \tan\theta$ for the ft/NM values, and this is the derivation of the values in the Vertical Path Angle box outlined with a heavy border. Outside this box, the climb rates are rounded to the nearest multiple of 5. The rest of the table values use the $\delta_{VSI}$ formula above.

For reference, see a Google spreadsheet that computes climb/descent rates using the above. The formula for altitude change for a climb or descent at 2.0° over 1 NM is of the form

=mround($O$1*tan(radians(B4)),5)


The value in O1 is the number of feet in one nautical mile. The trigonometric functions in Excel and Google Sheets deal in radians rather than degrees, where a complete circle has $2\pi$ radians.

The formula for the $\delta_{VSI}$ values in feet-per-minute is either

=mround(D$3*$C4/60,5)


or

=D$3*$C6/60


depending on whether the cell is inside the Vertical Path Angle box, i.e., whether the value should be rounded to the nearest multiple of 5 or not.

There are discrepancies, no more than 15 feet per minute. Some of the FAA figures are more conservative, some less. I was not able to discern a pattern; for details, see the color coding in the Computed sheet or the Deltas sheet in the linked workbook. The odd duck is for a 2.0° climb at 210-knot ground speed: the FAA table gives an unrounded value of 743 feet per minute rather than rounding as with its neighbors.

Stack Exchange does not support tables in their Markdown flavor, so a screenshot of the computed table is below. 