Drain Slope and Pipe Fall Calculator (Inches per Foot, 1/n, Percent)

Choose what to find, then enter the pipe run and the slope (or the fall). Sections 2 and 3, the pipe size and the starting invert depth can be left blank.

Units

What to find

Enter the slope as

Section 1 pipe run

Slope

1 /

Section 2 run (junction to junction, optional)

Slope

1 /

Section 3 run (optional)

Slope

1 /

Drain pipe size (optional, to compare with a guideline)

Also find the invert depth (optional)

Invert depth at the start (below grade, optional)

Enter all lengths and falls as positive numbers greater than 0. You can choose the units for the length and the fall separately (they are converted for the calculation).

Result and figure

Enter the pipe run and the slope (or the fall) on the left and press "Calculate". The result will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the pipe run and the slope (in inches per foot such as 1/4 or 1/8 in/ft, a fraction such as 1/96, or a percent) and see the fall between the start and end of the pipe (the difference in invert height) in inches
For a line whose slope or length changes at catch basins or cleanouts, enter the length and slope of each section (up to 3) to get the fall of each section and the total fall. Enter the invert depth at the start to also get the depth at each junction and at the end
The other way around, enter the fall and the pipe run to get the actual slope (in/ft, 1/n and percent), and compare it with the minimum slope for the pipe size in the International Plumbing Code (1/4, 1/8 or 1/16 in/ft)
For floors such as showers, patios and garage slabs, find the fall from the distance to the drain and the slope, or the slope from the distance and the fall
A section drawing of the pipe (start, end, fall and slope; a bent line through the junctions for several sections) is drawn, and a plain explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are on this page too

This page finds the fall when you slope a drain pipe or a floor so water flows (drainage slope). To convert a slope itself between percent, degrees, pitch and ratio, use the slope calculator; to size a ramp that people or wheelchairs go up, use the ramp length calculator (both are in the related pages). The pipe size guidelines here are common values from model codes and do not check compliance with your local plumbing code. Surveying the grade of land itself is not covered. Switch "Units" above the calculator to Metric to work in meters and millimeters.

What is this calculation used for?

Burying a yard drain line yourself (downspout to outlet)

To run a drain pipe 30 ft from a downspout to a pop-up emitter or a street-side outlet at 1/8 in per foot, the fall is \(30 \times 1/8 = 3.75\) in. If the invert at the start is 12 in below grade, the end is 15.75 in deep, so before you dig you can check whether that depth works and whether the outlet is low enough.
If you turn through a catch basin along the way, enter the length and slope of each section to get the total fall. Connections to the public sewer and sewer lines usually need a licensed plumber and a permit, so DIY is limited to what your local rules allow, such as storm water and yard drains on your own property. Call 811 before you dig.

Checking whether you can extend a drain when moving a sink or washer

Moving a kitchen sink 6 ft calls for extending the drain under the floor by 6 ft. A 2 in pipe at the IPC minimum of 1/4 in per foot needs \(6 \times 1/4 = 1.5\) in of extra fall, and whether you have that much room under the floor (from the bottom of the floor down to the existing drain connection) decides whether the move works.
If there is not enough room, you can compare raising the floor, shortening the route, or using a 3 in pipe at a gentler minimum slope (1/8 in per foot) before you talk to a plumber. The actual pipe size, slope and trap arm limits depend on the fixtures and your local code, so confirm the final plan with a designer or plumber.

Sloping a shower floor, patio or garage slab to a drain

If the farthest point of a shower floor is 5 ft from the drain, a slope of 1/4 in per foot (the IRC minimum for showers) gives \(5 \times 1/4 = 1.25\) in of fall. For a garage slab 20 ft deep at 1/8 in per foot, it is \(20 \times 1/8 = 2.5\) in. Build this fall into the mortar bed or the slab, lower on the drain side.
Floors are not sloped as steeply as pipes, because people walk and set furniture on them. Some finishes hold water at a gentle slope (textured tile), and some spots feel tilted at a steep slope, so check the finish maker's instructions and examples when you decide.

Finding why a drain is slow or clogs often (checking an existing line)

Measure the invert height at both ends of a slow drain and divide by the run to get the actual slope. If a 24 ft run has only 1.5 in of fall, that is 1.5 ÷ 24 = 1/16 in per foot (1/192), gentler than the IPC minimum of 1/8 in per foot for a 3 or 4 in pipe. A section with negative fall (back pitch) or a belly holds water and causes clogs.
If the slope is the cause, cleaning alone will not fix it for long. Relaying a pipe is costly, so use this calculation to see whether slope is the problem before you call a plumber.

A French drain or channel drain for a soggy lawn or garden

For a yard that drains poorly, people bury a French drain (perforated pipe in gravel) or install a channel drain. A 50 ft run at 1/8 in per foot (about 1%, a common minimum for French drains) falls 6.25 in, and at 1/4 in per foot 12.5 in, so the outlet (a dry well, a daylight outlet or a curb) must be at least that much lower.
The outlet height often cannot change, so first check what slope you can get over the run to the outlet, then decide the route and depth. Surround the buried pipe with gravel and fabric to keep it from clogging, and check with a string line as you work that the slope never reverses.

Seeing ratios and proportions in real life (middle school math)

"At 1/8 in per foot, a 20 ft pipe drops 2.5 in" is ratios and proportional relationships in action (Grades 6–7). Double the run and the fall doubles; double the slope denominator (1/48 → 1/96) and the fall is halved. These relationships show up as real dimensions on a job.
That in/ft, fractions, percents and ‰ are different ways to write the same slope is also a real example of unit rates and percents. Check how the "ratios" in a textbook set the dimensions of the pipes under a house and in a yard.

Formulas and figures

Pipe fall (run ÷ slope denominator)

Figure

Standard notation (the usual math form)

\(h\) \(=\) \(L\) \(\div\) \(n\)

In words (symbols replaced with words)

③ \(h\): fall \(=\) ① \(L\): pipe run \(\div\) ② \(n\): slope denominator (the \(n\) in slope \(1/n\))

The formula in words

① Divide the \(L\): pipe run

② by the \(n\): slope denominator

③ and you get the \(h\): fall (how much lower the invert is at the end than at the start)

Quick example

For a 20 ft (240 in) drain pipe sloped at 1/96 (1/8 in per foot), the fall from start to end is

fall \(h\) \(=\) pipe run (240 in) \(\div\) slope denominator (96)

\(240 \div 96 = 2.5\ (\mathrm{in})\)

Key idea

A slope of \(1/n\) drops 1 for every \(n\) across, so the pipe run divided by \(n\) is the fall from start to end. The larger \(n\), the gentler the slope: the same 20 ft pipe falls 5 in at 1/48 and 1.25 in at 1/192. In US plumbing, slope is usually written in inches per foot. One foot is 12 in, so 1/4 in/ft is 1/48, 1/8 in/ft is 1/96 and 1/16 in/ft is 1/192. With in/ft, the fall is simply the run in feet × the slope: \(20 \times 1/8 = 2.5\) in, the same answer. With a percent slope, \(h = L \times p \div 100\); with per mille (‰), \(h = L \times q \div 1000\). 1%, 10‰ and 1/100 are the same slope written in different ways. Use the same unit for the run and the fall in \(h = L \div n\) (240 in ÷ 96 = 2.5 in). If you divide feet by 96 and read the answer as inches, it is 12 times off. This calculator converts the units for you, so you can show the run in feet and the fall in inches. The "pipe run" here is the horizontal length. For gentle slopes like these, the difference from the length along the pipe itself is only about 0.005%, so in practice they are treated the same.

The actual slope from the fall and the run

Figure

Standard notation (the usual math form)

\(n\) \(=\) \(L\) \(\div\) \(h\)

\(p\) \(=\) \(h\) \(\div\) \(L\) \(\times\) \(100\)

\(q\) \(=\) \(h\) \(\div\) \(L\) \(\times\) \(1000\)

In words (symbols replaced with words)

③ \(n\): slope denominator (the \(n\) in slope \(1/n\)) \(=\) ① \(L\): pipe run \(\div\) ② \(h\): fall

⑤ \(p\): percent slope \(=\) \(h\): fall \(\div\) \(L\): pipe run \(\times\) ④ 100 (per 100 across)

⑦ \(q\): per mille slope \(=\) \(h\): fall \(\div\) \(L\): pipe run \(\times\) ⑥ 1000 (per 1000 across)

The formula in words

① Divide the \(L\): pipe run

② by the \(h\): fall

③ to get the \(n\): slope denominator (the slope is \(1/n\); the larger \(n\), the gentler)

④ The other way around, divide the fall \(h\) by the run \(L\) and multiply by 100

⑤ to get the \(p\): percent slope

⑥ In the same way, divide \(h\) by \(L\) and multiply by 1000

⑦ to get the \(q\): per mille slope

Quick example

For a 24 ft (288 in) pipe where you can only get 1.5 in of fall from start to end, the slope is

slope denominator \(n\) \(=\) pipe run (288 in) \(\div\) fall (1.5 in)

\(n = 288 \div 1.5 = 192 \quad \longrightarrow \quad \dfrac{1}{192} = \dfrac{1}{16}\ \mathrm{in/ft}\)

\(p = 1.5 \div 288 \times 100 \approx 0.52\ (\%)\)

\(q = 1.5 \div 288 \times 1000 \approx 5.2\ (‰)\)

Key idea

This formula checks "I can only get this much fall; what slope is that?". The run divided by the fall is the denominator of the fraction \(1/n\). Compare it with the guideline, and you see right away whether the slope is steeper or gentler. In in/ft, divide the fall in inches by the run in feet: 1.5 ÷ 24 = 1/16 in/ft. The example's \(1/192\) (1/16 in/ft) is gentler than the IPC minimum of 1/8 in/ft (1/96) for a 3 or 4 in pipe (the larger the denominator, the gentler). A slope that is too gentle slows the water, so dirt and solids stay in the pipe and cause clogs. A slope that is far too steep is also said to let water run ahead and leave solids behind, so very steep slopes are usually avoided. If the result is gentler than the guideline, consider a shorter run (a different route), a higher start, or a deeper end (the catch basin or the connection to the sewer). A percent slope is "how much it drops for every 100 across", and per mille is "for every 1000 across"; they are the same slope shown in different ways. Remember 1/4 in/ft = 1/48 ≈ 2.08%, 1/8 in/ft = 1/96 ≈ 1.04% and 1/16 in/ft = 1/192 ≈ 0.52%, and you can read drawings and specs written either way.

Total fall over several sections through junctions

Figure

Standard notation (the usual math form)

\(H\) \(=\) \(h_{1}\) \(+\) \(h_{2}\) \(+\) \(h_{3}\)

\(h_{i}\) \(=\) \(L_{i}\) \(\div\) \(n_{i}\)

In words (symbols replaced with words)

④ \(H\): total fall \(=\) \(h_{1}\): section 1 fall \(+\) \(h_{2}\): section 2 fall \(+\) \(h_{3}\): section 3 fall

③ \(h_{i}\): fall of section \(i\) \(=\) ① \(L_{i}\): run of section \(i\) \(\div\) ② \(n_{i}\): slope denominator of section \(i\)

The formula in words

① For each section, divide the \(L_{i}\): run of the section

② by the \(n_{i}\): slope denominator of the section

③ to get the \(h_{i}\): fall of the section and add them all up to get the

④ \(H\): total fall (the difference between the start and end inverts)

Quick example

For a line through 2 junctions, with section 1 at 16 ft and 1/8 in/ft (1/96), section 2 at 12 ft and 1/8 in/ft, and section 3 at 20 ft and 1/4 in/ft (1/48), the total fall is

total fall \(H\) \(=\) 192 in ÷ 96 \(+\) 144 in ÷ 96 \(+\) 240 in ÷ 48

\(H = 2 + 1.5 + 5 = 8.5\ (\mathrm{in})\)

Key idea

A drain line is split into sections at junctions such as catch basins and cleanouts, and the direction or slope can change at each one. The fall of each section is "run ÷ slope denominator" as in the first formula (or run in feet × in/ft), and the fall from start to end is their total. Water just drops inside a catch basin, so its inside width is not added to the run here. Drawings often measure slope center to center of the basins; in that case include that length in the section run (the fall comes out a little larger). If every section has the same slope, the answer is the same as "total run ÷ denominator" (16 + 12 + 20 = 48 ft, all at 1/8 in/ft, gives 6 in). Calculating section by section matters when, as in the example, the slope changes where lines join, pipe sizes change, or one section needs to drain faster. The total fall directly tells you how deep the end must be. If the height at the end is fixed, such as the connection to the public sewer or an existing catch basin, work backward from it to set the start height and the length of the route.

Invert depth at the end (burial depth)

Figure

Standard notation (the usual math form)

\(D\) \(=\) \(D_{0}\) \(+\) \(H\)

In words (symbols replaced with words)

③ \(D\): invert depth at the end \(=\) ① \(D_{0}\): invert depth at the start \(+\) ② \(H\): total fall

The formula in words

① Add the \(D_{0}\): invert depth at the start

② and the \(H\): total fall

③ to get the \(D\): invert depth at the end (from the ground surface to the invert at the end; the depth at each junction works the same way with the fall up to that point)

Quick example

If the invert at the start is 12 in below grade and the total fall to the end is 8.5 in, the invert depth at the end is

invert depth at the end \(D\) \(=\) start depth (12 in) \(+\) total fall (8.5 in)

\(D = 12 + 8.5 = 20.5\ (\mathrm{in})\)

Key idea

Pipe heights are measured at the bottom inside of the pipe (the invert), not at the center or the top, and depths are given from the ground surface (grade), such as "12 in below grade". A downhill slope makes the invert deeper toward the end, so the start depth plus the total fall is the end depth. Once you know the depth, you can check how deep to dig (add the pipe's outside diameter and the bedding below it), whether the end reaches the height of the catch basin or the sewer connection, and whether the pipe runs too shallow anywhere (under driveways, or where the ground freezes, a shallow pipe can be damaged). Follow local conditions and code for trench depth and pipe protection, and call 811 before you dig. If the ground is not level (the lot slopes), the "depth" in this formula is measured from the ground at the start. If the ground at the end is at a different height, add or subtract that difference (surveying the grade of the land itself is not covered on this page).

The fall of a pipe is "run ÷ slope denominator \(n\)" (\(h = L \div n\)), or in US terms "run in feet × inches per foot": a 20 ft pipe at 1/8 in/ft (1/96) falls 2.5 in. To find the slope from the fall and the run, use \(n = L \div h\) (or fall ÷ run in feet for in/ft) and compare it with a guideline such as the IPC minimum of 1/8 in/ft; the larger the denominator, the gentler. For a line through junctions, add up the fall of each section, and add that to the start invert depth to get the end invert depth. Floor slopes to a drain use exactly the same formulas.

Symbols and terms

Symbols

\(h\)	aitch	The fall, from "height": how much lower the invert is at the end than at the start. It is found with \(h = L \div n\).
\(L\)	ell	The pipe run (for a floor, the distance to the drain), from "length". It is the horizontal length; for gentle slopes, the difference from the length along the pipe can be ignored.
\(n\)	en	The denominator when the slope is written as the fraction \(1/n\): a slope that drops 1 for every \(n\) across, with \(n = L \div h\) (1/8 in/ft is \(n = 96\)). The larger \(n\), the gentler the slope.
\(p\)	pee	The percent slope, from "percent". \(p = h \div L \times 100\), how much it drops for every 100 across (1/96 is about 1.04%).
\(q\)	cue	The per mille slope (‰), using the letter after \(p\). \(q = h \div L \times 1000\), how much it drops for every 1000 across (1/100 is 10‰).
\(H\)	capital aitch	The total fall over several sections through junctions: the falls of each section \(h_{1}, h_{2}, h_{3}\) added together, the difference between the start and end inverts.
\(h_{i},\ L_{i},\ n_{i}\)	aitch sub i, ell sub i, en sub i	The fall, run and slope denominator of section \(i\) (1, 2, 3). The small \(i\) at the lower right is a subscript for the section number.
\(D_{0}\)	dee sub zero	The invert depth at the start (below grade), from "depth" with 0 for the starting point.
\(D\)	dee	The invert depth at the end (below grade). It is found with \(D = D_{0} + H\).
\(\sum\)	sigma	The summation sign, the Greek letter sigma (the Greek S, for "sum"). \(\sum_{i=1}^{3} h_{i}\) is "add up \(h_{1}\) through \(h_{3}\)".
‰	per mille	A unit for parts per thousand, one tenth of a percent: 10‰ = 1%. Some sewer and civil drawings use it for slopes.

Terms

Drainage slope	The gentle downhill slope given to drain pipes and channels so water flows to the low end on its own. In the US it is written in inches per foot (1/4 in/ft, 1/8 in/ft). Too gentle and the pipe drains poorly and clogs; too steep is said to leave solids behind, so it is set within a range for the pipe size.
Fall	How much lower the end is than the start, for a pipe, a floor or the ground. Plumbers say "1/4 inch of fall per foot". Floors such as showers and patios are also sloped this way, often written in in/ft or percent.
Slope	How much something rises or drops compared with the distance across: fall ÷ horizontal run. For drains it is written in in/ft in the US, and the same slope can be written as a fraction (1/96), a percent (%) or per mille (‰).
Fractional slope	Writing a slope that drops 1 for every \(n\) across as \(1/n\), such as 1/96 for a drain or 1:12 for a ramp. The larger \(n\), the gentler, and the fall is the run divided by \(n\).
Percent slope	A slope written as how much it drops for every 100 across: fall ÷ run × 100. 1/100 is 1%, 1/8 in/ft is about 1.04% and 1/4 in/ft is about 2.08%.
Per mille	A unit for parts per thousand (‰). For slopes it is how much it drops for every 1000 across; 10‰ is the same as 1% and 1/100. It appears on some sewer, road and railway drawings.
Invert	The bottom of the inside of a drain pipe. Pipe heights and depths are measured at the invert, not at the center or top of the pipe; "invert elevation" on drawings refers to this point. Water flows along it.
Catch basin (junction)	A box on a drain line for inspection and cleaning, placed where the line changes direction or slope, where lines join, or partway along a long run. In buildings and sewer laterals, a cleanout plays a similar role. The pipe is split into sections there, so the slope is calculated section by section.
Nominal pipe size	The standard number for a pipe's size, such as 2, 3 or 4 in for PVC or ABS drain pipe, close to the inside diameter. Minimum drain slopes are set by pipe size, and smaller pipes need steeper slopes.
Burial depth	The depth from the ground surface to the invert of a buried pipe. With a downhill slope, it gets deeper downstream. Too shallow and a pipe can be damaged by vehicle loads or frost, so keep enough depth for the location.
Grade (ground surface)	The finished ground surface used as the reference for heights around a building. Pipe depths are given from it, such as "12 in below grade".
Slab	A concrete floor on the ground, such as a garage, patio, driveway or basement floor. It is usually finished with a slope toward a drain or the outside (often about 1/8 to 1/4 in/ft) so rain and wash water do not pool.
Back pitch (belly)	A slope that runs the wrong way, with the downstream end higher; a sag in the middle of a pipe is called a belly. Water sits there, flow slows and clogs form. When an existing drain acts up, first check whether any part slopes the wrong way.
Gravity drainage	Letting water flow to a lower point by slope alone, without a pump. Home drains are basically gravity drainage, so every section needs a downhill slope.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.

Ratios, unit rates and percents (Grade 6)	A slope (fall ÷ run) is a kind of ratio, and "inches per foot" is a unit rate Going between decimals and percents, such as 0.01 = 1% and 0.005 = 0.5%
Fractions (Grades 4–5)	Reading the denominator 96 in \(\dfrac{1}{96}\) as the ratio "1 of fall for every 96 across" "Multiplying by \(\dfrac{1}{96}\)" is the same as "dividing by 96", and 1/8 in per foot is 1 in per 8 ft
Converting units of length (Grade 4)	1 ft = 12 in, so you can put the run and the fall in the same unit (20 ft = 240 in)
Proportional relationships (Grade 7)	At the same slope, doubling the run doubles the fall (the fall is proportional to the run)
Expressions and subscripts (Grade 6 and up)	The small numbers at the lower right of \(h_{1}, h_{2}, h_{3}\) are subscripts for the section number \(\sum\) (sigma) is a sign for "add them all up" (the calculation itself is just addition)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the pipe fall

Pipe run L (ft)	20
Slope (in/ft)	0.125
Fall h (in)	=B1*B2
Slope denominator n (slope 1/n)	=12/B2

Table to find the actual slope from the fall and run

Pipe run L (ft)	24
Fall h (in)	1.5
Slope (in/ft)	=B2/B1
Slope denominator n (slope 1/n)	=B1*12/B2
Percent slope p (%)	=B2/(B112)100

Table to find the total fall over several sections

Section 1 run L1 (ft)	16
Section 1 slope (in/ft)	0.125
Section 2 run L2 (ft)	12
Section 2 slope (in/ft)	0.125
Section 3 run L3 (ft)	20
Section 3 slope (in/ft)	0.25
Total fall H (in)	=B1B2+B3B4+B5*B6

Table to find the invert depth at the end

Invert depth at the start D0 (in)	12
Total fall H (in)	8.5
Invert depth at the end D (in)	=B1+B2

After pasting, the upper rows of column B are the inputs and the last rows are calculated automatically.
In the first table B3 is 2.5 and B4 is 96. In the second, B3 to B5 are 0.0625, 192 and about 0.52. In the third, B7 is 8.5, and in the fourth, B3 is 20.5.
These tables use the US style of run in feet × slope in inches per foot. To use a fraction 1/n instead, put the run and the fall in the same unit (inches) and use "=B1/B2" with n in B2. For a percent slope, fall = run × percent ÷ 100 (in the same unit).

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the pipe fall

Pipe run L (ft)	20
Slope (in/ft)	0.125
Fall h (in)	=B1*B2
Slope denominator n (slope 1/n)	=12/B2

Table to find the actual slope from the fall and run

Pipe run L (ft)	24
Fall h (in)	1.5
Slope (in/ft)	=B2/B1
Slope denominator n (slope 1/n)	=B1*12/B2
Percent slope p (%)	=B2/(B112)100

Table to find the total fall over several sections

Section 1 run L1 (ft)	16
Section 1 slope (in/ft)	0.125
Section 2 run L2 (ft)	12
Section 2 slope (in/ft)	0.125
Section 3 run L3 (ft)	20
Section 3 slope (in/ft)	0.25
Total fall H (in)	=B1B2+B3B4+B5*B6

Table to find the invert depth at the end

Invert depth at the start D0 (in)	12
Total fall H (in)	8.5
Invert depth at the end D (in)	=B1+B2

The same formulas as Excel work as is (they use only basic arithmetic, so there are no function name differences). Copy the whole table, paste it into cell A1 and change column B to your own numbers.

How to calculate it in Python

run_ft = 20           # pipe run L (ft)
slope_in_per_ft = 1/8 # slope (in/ft). 1/4 in/ft -> 0.25, 1/8 in/ft -> 0.125

drop_in = run_ft * slope_in_per_ft               # fall h (in)
denominator = 12 / slope_in_per_ft               # slope as 1/n
percent = slope_in_per_ft / 12 * 100             # percent slope (%)
print(f"Fall: {drop_in:.2f} in  (slope {slope_in_per_ft} in/ft = 1/{denominator:.0f} = {percent:.2f} %)")

# The other way around: actual slope from the fall and run, compared with the IPC minimum (1/8 in/ft for 3-6 in pipe)
run2_ft = 24          # pipe run (ft)
drop2_in = 1.5        # fall you can get (in)
actual = drop2_in / run2_ft
minimum = 1/8
needed_drop = run2_ft * minimum
print(f"Actual slope: {actual:.4f} in/ft (1/{run2_ft * 12 / drop2_in:.0f}); fall needed for 1/8 in/ft is {needed_drop:.2f} in, difference {drop2_in - needed_drop:.2f} in (negative = short)")

# Total fall over several sections and the invert depth at the end
sections = [(16, 1/8), (12, 1/8), (20, 1/4)]      # (run ft, slope in/ft) for each section
start_depth_in = 12                               # invert depth at the start (in below grade)
total_drop = 0
for i, (section_run, section_slope) in enumerate(sections, start=1):
    section_drop = section_run * section_slope
    total_drop += section_drop
    print(f"Section {i}: {section_run} ft x {section_slope} in/ft = {section_drop:.2f} in (total so far {total_drop:.2f} in, invert {start_depth_in + total_drop:.2f} in below grade)")
print(f"Total fall: {total_drop:.2f} in, invert depth at the end: {start_depth_in + total_drop:.2f} in below grade")

It runs with the standard library only (basic arithmetic). Change the run and slope at the top to your own numbers and run it. Runs are in feet and slopes in inches per foot, so falls come out in inches. Add or remove tuples in sections to change the number of sections.

How to write it in LaTeX and other math languages (copy and paste)

Pipe fall (run ÷ slope denominator)

h = L ÷ n

h = \dfrac{L}{n}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>h</mi>
    <mo>=</mo>
    <mfrac><mi>L</mi><mi>n</mi></mfrac>
  </mrow>
</math>

h = L / n

L/n

h := L/n;

h = L/n;

h = L/n

The actual slope from the fall and the run

n = L ÷ h,  p = h ÷ L × 100,  q = h ÷ L × 1000

n = \frac{L}{h},\quad p = \frac{h}{L} \times 100,\quad q = \frac{h}{L} \times 1000

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>n</mi><mo>=</mo><mfrac><mi>L</mi><mi>h</mi></mfrac>
    <mo>,</mo>
    <mi>p</mi><mo>=</mo><mfrac><mi>h</mi><mi>L</mi></mfrac><mo>×</mo><mn>100</mn>
    <mo>,</mo>
    <mi>q</mi><mo>=</mo><mfrac><mi>h</mi><mi>L</mi></mfrac><mo>×</mo><mn>1000</mn>
  </mrow>
</math>

n = L / h,  p = h / L * 100,  q = h / L * 1000

{L/h, h/L*100, h/L*1000}

n := L/h;  p := h/L*100;  q := h/L*1000;

n = L/h; p = h/L*100; q = h/L*1000;

n = L/h, p = h/L × 100, q = h/L × 1000

Total fall over several sections through junctions

H = h₁ + h₂ + h₃,  hᵢ = Lᵢ ÷ nᵢ

H = \sum_{i=1}^{3} h_{i},\quad h_{i} = \dfrac{L_{i}}{n_{i}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>H</mi><mo>=</mo>
    <munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></munderover>
    <msub><mi>h</mi><mi>i</mi></msub>
    <mo>,</mo>
    <msub><mi>h</mi><mi>i</mi></msub><mo>=</mo>
    <mfrac><msub><mi>L</mi><mi>i</mi></msub><msub><mi>n</mi><mi>i</mi></msub></mfrac>
  </mrow>
</math>

H = sum_(i=1)^3 h_i,  h_i = L_i / n_i

Sum[L[i]/n[i], {i, 1, 3}]

H := sum(L[i]/n[i], i = 1..3);

H = sum(L ./ n);

H = ∑_(i=1)^3 h_i, h_i = L_i/n_i

Invert depth at the end (burial depth)

D = D₀ + H

D = D_{0} + H

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>D</mi><mo>=</mo><msub><mi>D</mi><mn>0</mn></msub><mo>+</mo><mi>H</mi>
  </mrow>
</math>

D = D_0 + H

D0 + H

Dend := D0 + H;

D = D0 + H;

D = D_0 + H

How to have ChatGPT do the calculation

You are an assistant for plumbing and site drainage calculations. Do the calculations below by actually running Python code, and base your answer only on the numbers from the output (do not calculate in your head or guess).

I am laying an outdoor drain line through 2 catch basins. Section 1 is 16 ft at 1/8 in per foot, section 2 is 12 ft at 1/8 in per foot, and section 3 is 20 ft at 1/4 in per foot. The invert at the start is 12 in below grade.
Find each of the following (in inches).
1. The fall of each section (run in feet × slope in inches per foot)
2. The total fall from start to end
3. The invert depth at catch basin 1, catch basin 2 and the end (start depth + fall up to that point)
4. The fall if the whole 48 ft run were at a uniform 1/8 in per foot, and how it differs from the end depth in step 3

Show the formulas you used and the numbers from the output.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula