Ramp Length Calculator (Rise and Slope, ADA 1:12)

Choose what you want to find, then enter the rise and the slope (or the length you have). Landings, width and price can be left blank.

Units

What to find

Rise (height of the step)

Enter the slope as

Slope ratio n (the n in 1:n)

1 :

Include landings (optional)

Length of one landing (optional)

Number of landings

Also find the area and cost (optional)

Ramp width (optional)

Unit price ($, optional)

Enter all lengths as positive numbers. You can choose different units for the rise and for the lengths (they are converted before calculating).

Result and figure

Enter the rise and the slope (or the length you have) in the fields 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 rise (the height of the step) and the slope (a ratio such as 1:12 or 1:15, a percent, or an angle), and you get the horizontal run the ramp needs and the length along the slope (the length of the ramp itself)
Or enter the rise and the length you have room for, and you get the actual slope (1:n, % and degrees) and a table of the length needed at common slopes (1:8, 1:10, 1:12, 1:15 and 1:20)
When you choose a ready-made threshold ramp (rubber, plastic or aluminum), enter the rise and the length of the product, and you get the slope it will have
You can also get the total length with landings (the flat parts), the area from the ramp width, and an estimated cost if you enter a price (optional)
The page draws the right triangle of rise, run and slope with a line for the slope you compare against. A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are also on this page

This page finds how much length a ramp needs for a given rise. To convert a slope between percent, degrees and ratio, use the "Slope Calculator". To plan stairs instead, use the "Stair Calculator" (both are in the related pages). The slopes shown here are common guidelines. This page does not check compliance with the ADA Standards, building codes or local rules.

What is this calculation used for?

Adding a wheelchair ramp at the front porch (home accessibility)

If the porch is 24 in above the ground, a 1:12 ramp needs $24 \times 12 = 288$ in (24 ft) of horizontal run, and a 1:15 ramp needs 360 in (30 ft). Before you talk to a contractor, you can check whether the yard has that much room, or whether you need a switchback ramp that turns around at a landing.
A person who pushes their own wheelchair needs a gentler slope than one who is pushed, and 1:12 to 1:15 is a common guide, but it also depends on the user's strength and on whether the chair is powered. For a ramp at home, check the local building code and whether a permit is needed.

Choosing a threshold ramp online (rubber, plastic or aluminum)

Threshold ramps are sold by rise, such as "for a 2 in rise, 12 in long". The length ÷ the rise gives $n$: here $12 \div 2 = 6$, so the slope is 1:6 (about 9.5°), which is quite steep.
For a small step inside the house, a short product can be fine, but if the wheelchair user pushes on their own, this calculation shows that a longer product is better (a 1:12 slope over the same 2 in rise needs a 24 in long ramp). Follow the maker's listed rise range and weight limit.

Making a store or office entrance accessible for wheelchairs and strollers

If there is a 6 in step from the sidewalk to the door, a 1:12 ramp needs 72 in (6 ft) of horizontal run and a 1:8 ramp needs 48 in (4 ft). Working out first how much room there is in front of the door (without blocking the sidewalk) helps you decide between a permanent ramp and a portable one you set out when needed.
A customer entrance must serve many people, including self-propelled wheelchair users, strollers and older adults, and in the US it falls under the ADA Standards (1:12 or gentler). For existing buildings where space is tight, the ADA allows a steeper slope only for a very small rise (1:10 for a rise of 6 in or less, 1:8 for 3 in or less). Check with a designer or the local building department before building a permanent ramp.

Making a temporary ramp for a hand truck when moving or loading (DIY and events)

To roll a hand truck or equipment cases over a step at a loading door, people often lay a plywood board or aluminum ramp across it. Lay an 8 ft (96 in) board over a 10 in step and the horizontal run is $\sqrt{96^{2} - 10^{2}} \approx 95.5$ in, a slope of about 1:9.5 (about 6.0°).
A heavily loaded hand truck takes a lot of force to push up a steep slope, and on the way down it can pull away from you, which is dangerous. For a hand truck pushed by hand, 1:8 to 1:10 is common, and heavy or powered carts need a gentler slope. Also check that the board does not bend too much and is not slippery.

Building a ramp to roll a bike or motorcycle into a garage or shed

A ramp over a garage or shed step (about 6 to 8 in high) lets you roll a bicycle or motorcycle in and out without lifting it. When a person walks the bike up, slopes of about 1:6 to 1:8 are used. For a 7 in step at 1:8, the horizontal run is 56 in and the board along the slope is about 56.4 in.
The board (the length along the slope) is a little longer than the horizontal run, so check the "length along the slope" on this page before buying lumber. For a heavy motorcycle, the strength of the board and a non-slip surface also matter.

Making a pet ramp to help an older dog or cat

Older dogs and cats, and short-legged breeds, can hurt their joints jumping on and off a sofa, a bed or the back of a car. For an 18 in high sofa with a 1:3 ramp (about 18.4°), the horizontal run is $18 \times 3 = 54$ in and the board is about 56.9 in long.
How steep a ramp an animal will climb varies, and small or older animals do better with a gentler slope. For the same height, you decide by trying different values of $n$, balancing "how long can the ramp be" against "how steep will my pet still climb".

Seeing where the Pythagorean theorem and inverse trig are used (middle and high school math)

Finding the board length from the horizontal run and the rise is exactly the Pythagorean theorem from 8th grade math. Finding "how many degrees is a 1:12 slope" uses $\tan$ backward (the inverse trig function $\arctan$) from high school geometry and precalculus, and it is one of the most familiar places it shows up.
Try connecting the textbook formulas to real measurements, such as a ramp at the front door or a board for moving heavy things.

Formulas and figures

Horizontal run needed (rise × ratio n)

Figure

Standard notation (the usual math form)

$L$ $=$ $h$ $\times$ $n$

In words (symbols replaced with words)

③ $L$: horizontal run needed $=$ ① $h$: rise $\times$ ② $n$: slope ratio (the $n$ in 1:$n$)

The formula in words

① Take the $h$: rise

② multiply it by the $n$: slope ratio

③ and you get the $L$: horizontal run needed (the length seen from above, from the edge of the step to the end of the ramp)

Quick example

The horizontal run needed to climb a 6 in step with a 1:12 ramp is

$L$: horizontal run needed $=$ rise (6 in) $\times$ slope ratio (12)

$6 \times 12 = 72\ (\mathrm{in})$

Key idea

Ramp slopes are usually written as a ratio with 1 in front, such as 1:12 or 1:15. "1:12" means "1 unit of rise for every 12 units of horizontal run", so multiplying the rise by 12 gives the horizontal run needed. The larger $n$ is, the gentler the slope and the more room you need (for the same 6 in rise, 1:15 needs 90 in and 1:8 needs 48 in). If the slope is given as a percent, use $L = h \div (p \div 100)$. If it is given as an angle, use $L = h \div \tan\theta$. Both are "rise ÷ slope", which is the same as multiplying by the reciprocal of the slope $i = 1/n$. Use the same unit for the rise and the length (6 in × 12 = 72 in, which is 6 ft). If you enter the rise in inches and then read the answer as feet, you are off by a factor of 12. This calculator converts units for you, so you can enter the rise in inches and the lengths in feet.

Length along the slope (length of the ramp itself)

Figure

Standard notation (the usual math form)

$s$ $=$ $\sqrt{L^{2} + h^{2}}$

$L$ $=$ $\sqrt{s^{2} - h^{2}}$

In words (symbols replaced with words)

② $s$: length along the slope $=$ ① square root of ($L$ squared + $h$ squared)

④ $L$: horizontal run $=$ ③ square root of ($s$ squared − $h$ squared)

The formula in words

① Take the square root of ($L$ squared + $h$ squared) (the Pythagorean theorem)

② and you get the $s$: length along the slope (the length of the ramp board or surface itself)

③ Going the other way, when the board length is fixed, take the square root of ($s$ squared − $h$ squared)

④ and you get the $L$: horizontal run (the length that board covers, seen from above)

Quick example

For a ramp with a 6 in rise and a 72 in horizontal run (1:12), the length along the slope is

$s$: length along the slope $=$ square root of (72 in squared + 6 in squared)

$s = \sqrt{72^{2} + 6^{2}} = \sqrt{5184 + 36} = \sqrt{5220} \approx 72.25\ (\mathrm{in})$

Key idea

The rise $h$, the horizontal run $L$ and the length along the slope $s$ are the three sides of a right triangle, so $s$, the hypotenuse, comes from the Pythagorean theorem. The ramp board, a ready-made ramp or a concrete surface is as long as $s$, so this is the length you need in materials. With a gentle slope such as 1:12, the length along the slope is only slightly longer than the horizontal run (about 0.35% at 1:12: 72.25 in against 72 in in the example). The steeper the slope, the bigger the difference: about 0.8% at 1:8, about 3% at 1:4 and about 12% at 1:2. When the length along the slope is fixed first, as in "I have an 8 ft board", use the second formula to turn it into a horizontal run, then find the slope. If the board is shorter than the rise, no right triangle can be made (the number under the square root is negative), and this calculator shows an error.

Actual slope from the length you have

Figure

Standard notation (the usual math form)

$n$ $=$ $L$ $\div$ $h$

$p$ $=$ $h$ $\div$ $L$ $\times$ $100$

$\theta$ $=$ $\arctan$ $($ $h$ $\div$ $L$ $)$

In words (symbols replaced with words)

③ $n$: slope ratio (the $n$ in 1:$n$) $=$ ① $L$: horizontal run $\div$ ② $h$: rise

⑤ $p$: percent slope $=$ $h$: rise $\div$ $L$: horizontal run $\times$ ④ 100 (per 100 of horizontal run)

⑦ $\theta$: angle $=$ $\arctan$ $($ ⑥ $h$: rise $\div$ $L$: horizontal run $)$

The formula in words

① Take the $L$: horizontal run

② divide it by the $h$: rise

③ and you get the $n$: slope ratio (the slope is 1:$n$, and a larger $n$ is gentler)

④ Going the other way, divide the rise $h$ by the horizontal run $L$ and multiply by 100

⑤ to get the $p$: percent slope

⑥ Put the $h$: rise divided by the horizontal run $L$ into the arctangent $\arctan$ (the inverse of $\tan$)

⑦ and you get the $\theta$: angle

Quick example

The slope when you have only 120 in (10 ft) of horizontal run for a 12 in rise is

$n$: slope ratio $=$ horizontal run (120 in) $\div$ rise (12 in)

$n = 120 \div 12 = 10 \quad \longrightarrow \quad 1:10$

$p = 12 \div 120 \times 100 = 10\ (\%)$

$\theta = \arctan(12 \div 120) = \arctan 0.1 \approx 5.71^\circ$

Key idea

This formula answers "I only have this much room. How steep will the ramp be?" The horizontal run divided by the rise gives the $n$ in 1:$n$. Compare it with a guideline slope ($n = 12$ for 1:12) to see at once whether your ramp is steeper or gentler. The 1:10 in the example is steeper than 1:12 (the smaller $n$ is, the steeper the slope). A 1:12 slope would need $12 \times 12 = 144$ in, so the space is 24 in short. The percent slope is "how much it rises for every 100 of horizontal run", and the angle is the tilt from the horizontal. They are only different ways of showing the same slope. Note that percent and angle are not proportional (10% is about 5.7°, but 100% is 45°). If the length you have is the length along the slope (a board length), first turn it into a horizontal run with the earlier formula $L = \sqrt{s^{2} - h^{2}}$, then use this formula.

Total length with landings

Standard notation (the usual math form)

$T$ $=$ $L$ $+$ $d$ $\times$ $m$

In words (symbols replaced with words)

④ $T$: total length $=$ ① $L$: horizontal run needed $+$ ② $d$: length of one landing $\times$ ③ $m$: number of landings

The formula in words

① Take the $L$: horizontal run needed

② add the $d$: length of one landing

③ times the $m$: number of landings

④ and you get the $T$: total length (the horizontal length the whole ramp takes up)

Quick example

A 36 in rise at 1:12 needs 432 in of horizontal run. The ADA Standards allow at most 30 in of rise per sloped section, so the ramp is split into two sections with one 60 in landing between them. The total length is

$T$: total length $=$ horizontal run needed (432 in) $+$ landing length (60 in) $\times$ number of landings (1)

$432 + 60 \times 1 = 492\ (\mathrm{in})$

Key idea

A landing is a flat area for resting or turning, and its height does not change. So it is not part of the slope calculation; you simply add it to the horizontal run $L$ of the sloped parts. In the example, 492 in is 41 ft. Adding a landing does not change the slope of the sloped parts, so a landing does not let you make the ramp shorter. The ADA Standards also require a landing at the top and bottom of every sloped section. To include those in the total, count them in the number of landings. On a switchback ramp (one that makes a U-turn), the sloped parts are split in two and run in opposite directions, so the depth of the site needed is roughly "one sloped part (half of the horizontal run) + the landing". $T$ in this formula is the total when everything is laid out in a straight line. Check the layout of a switchback ramp on a drawing.

Area and estimated cost (optional)

Standard notation (the usual math form)

$A$ $=$ $s$ $\times$ $w$

$C$ $=$ $s$ $\times$ $u$

In words (symbols replaced with words)

③ $A$: ramp area $=$ ① $s$: length along the slope $\times$ ② $w$: ramp width

⑥ $C$: estimated cost $=$ ④ $s$: length along the slope (or $A$: area) $\times$ ⑤ $u$: unit price (per ft, or per ft²)

The formula in words

① Take the $s$: length along the slope

② multiply it by the $w$: ramp width

③ and you get the $A$: ramp area (the area to cover or paint)

④ Take the $s$: length along the slope (or $A$: area)

⑤ multiply it by the $u$: unit price

⑥ and you get the $C$: estimated cost

Quick example

For a ramp 6.021 ft (72.25 in) long along the slope and 3 ft (36 in) wide, the area, and the estimated cost at an assumed $150 per foot, are

$A$: ramp area $=$ length along the slope (6.021 ft) $\times$ width (3 ft)

$C$: estimated cost $=$ length along the slope (6.021 ft) $\times$ unit price ($150 per ft)

$A = 6.021 \times 3 \approx 18.06\ (\mathrm{ft^2})$

$C = 6.021 \times 150 \approx \$903$

Key idea

For the area, multiply the width by the length along the slope, not by the horizontal run. Non-slip paint, tiles and ramp surfaces go on the sloped surface, so measuring along the slope matches the real amount (with a gentle slope the difference from the horizontal run is very small). The unit price can be for materials or for installation. Multiply a price "per ft" by the length along the slope, and a price "per ft²" by the area. Prices vary a lot by product, region and type of work, so use the numbers from a quote or a product page. This calculator has no built-in price and shows a cost only when you enter one.

The horizontal run a ramp needs is "rise × slope ratio $n$" ($L = h \times n$), and the length along the slope comes from the Pythagorean theorem, $s = \sqrt{L^{2} + h^{2}}$. To find the slope from the length you have, use $n = L \div h$ and compare $n$ with a guideline such as 1:12 to see whether it is steeper or gentler. Landings are simply added to the horizontal run, and the area is the length along the slope × the width.

Symbols and terms

Symbols

$h$	aitch	The rise (the difference in height), from the first letter of "height". It is the height the ramp has to climb, such as from the ground to a porch or over a door threshold.
$n$	en	The slope ratio: the $n$ when the slope is written 1:$n$. It is a slope that rises 1 for every $n$ of horizontal run, and $n = L \div h$ ($n = 12$ for 1:12). The larger $n$ is, the gentler the slope.
$L$	el	The horizontal run. This page uses the first letter of "length". It is the length of the ramp seen from above, from the edge of the step to the end of the ramp, and $L = h \times n$.
$s$	ess	The length along the slope (the true length). This page uses the first letter of "slope". It is the length of the ramp board or surface itself, found with $s = \sqrt{L^{2} + h^{2}}$.
$i$	eye	The slope as a number (rise ÷ horizontal run), from the first letter of "inclination". $i = 1/n$, and it has no unit.
$p$	pee	The percent slope, from the first letter of "percent". $p = h \div L \times 100$, which is how much the ramp rises for every 100 of horizontal run (1:12 is about 8.33%).
$\theta$	theta	The angle of the ramp from the horizontal. Theta is the Greek letter often used for angles. It is found with $\theta = \arctan(h \div L)$ (1:12 is about 4.76°).
$T$	tee	The total length with landings, from the first letter of "total". $T = L + d \times m$, the horizontal length the whole ramp takes up.
$d$	dee	The length of one landing. This page uses the first letter of "depth" (how deep the landing is).
$m$	em	The number of landings. This page uses it as a count (it is not the unit meter, m).
$w$	double-u	The ramp width, from the first letter of "width". It is used for the area $A = s \times w$.
$A$	ay	The ramp area, from the first letter of "area". It is the length along the slope × the width.
$C$	see	The estimated cost, from the first letter of "cost". It is the length along the slope (or the area) × the unit price.
$u$	you	The unit price, from the first letter of "unit price". Enter the price per ft, or per ft².
$\tan$	tangent	Tangent. In a right triangle it is height ÷ base. The slope $i$ is exactly $\tan\theta$.
$\arctan$	arctangent, inverse tangent ($\tan^{-1}$)	The inverse of tangent. It returns the angle whose $\tan$ is a given value, and is also written $\tan^{-1}$. It is used to find the angle from the slope.
$\sqrt{\ }$	square root	Square root: the positive number that gives this number when squared. It is used with the Pythagorean theorem to find the length along the slope (example: $\sqrt{25} = 5$).
$\approx$	approximately equal to	The sign for "approximately equal". It is used when a value that does not come out evenly, such as a square root or an $\arctan$, is rounded to a decimal.

Terms

ramp	A sloped surface that connects two levels instead of steps. It lets wheelchairs, strollers, hand trucks and bicycles get over a step. The gentler the slope, the easier it is to climb, but the more room it needs.
rise	The difference in height between two floors or ground levels, such as from the ground to a porch, over a door threshold, or from the sidewalk to a store entrance. This page writes it $h$. If there are several steps, the rise is the total height from the bottom to the top.
slope	How much a surface goes up (or down) for the horizontal distance it covers, found as rise ÷ horizontal run. Ramp slopes are usually written as a ratio such as 1:12 or 1:15, and the same slope can also be shown as a percent (%) or an angle (°).
slope ratio	Writing a slope that rises 1 for every $n$ of horizontal run as 1:$n$, as in "a 1:12 ramp". The larger $n$ is, the gentler the slope, and the horizontal run needed is $n$ times the rise. (Drain pipes are usually given in inches per foot instead, such as 1/4 in per ft.)
percent slope	A way of giving a slope as how much it rises for every 100 of horizontal run, also called percent grade. It is rise ÷ horizontal run × 100. 1:12 is about 8.33%, and 1:8 is 12.5%.
angle of slope	The angle of the sloped surface from the horizontal. It is the slope shown in degrees (°) and is found with $\arctan$. Percent slope and angle are not proportional (10% is about 5.7°, but 100% is 45°).
horizontal run	The length of the ramp seen from directly above, from the edge of the step to the end of the ramp. How much room the ramp takes up on the site is set by this length. It is always shorter than the length along the slope.
length along the slope	The length of the ramp board or surface itself, sometimes called the true length. This is the length you actually need in materials, and it is found from the horizontal run and the rise with the Pythagorean theorem.
landing	A flat area at the top, bottom or middle of a ramp or stairs, or where it turns, for resting or changing direction. Its height does not change, so it is not part of the slope calculation; it is added to the horizontal run for the total length. The ADA Standards require landings at least 60 in long.
threshold ramp	A ready-made ramp that you simply place over a door threshold, a small step or a store entrance. They are made of rubber, plastic or aluminum, and each size has a set length (the horizontal length when installed) for a certain rise. The length ÷ the rise is the $n$ of its slope.
ADA Standards	The 2010 ADA Standards for Accessible Design, the design rules under the Americans with Disabilities Act. For ramps at public and commercial buildings they require a slope of 1:12 or gentler, no more than 30 in of rise per sloped section, landings at least 60 in long, and a clear width of at least 36 in, among other things. Homes are usually not covered, but many people use the same numbers as a guide.
self-propelled	A wheelchair user moving by turning the wheels with their own arms, without a helper pushing. A self-propelled wheelchair needs a gentler slope than one pushed by a helper, so this is a key point when choosing a slope.
Pythagorean theorem	The rule that in a right triangle, base squared + height squared = hypotenuse squared. It is used to find the length along the slope from the horizontal run and the rise.
arctangent	The inverse of $\tan$ ($\arctan$). It answers "what angle has this value of $\tan$?" On a scientific calculator it is the $\tan^{-1}$ key, and in Excel it is the ATAN function.
unit price	The price for one unit of amount, such as per ft or per ft². On this page you can enter a material or installation price, and it is multiplied by the length along the slope or the area to give an estimated cost.

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 fastest way forward.

Ratios and percents (Grade 6)	Knowing that a slope (rise ÷ horizontal run) is a kind of ratio, "part ÷ base" Being able to go between decimals and percents, such as 0.0833 as about 8.33% and 0.1 as 10%
Fractions and reciprocals (Grades 5–6)	Reading the 12 in 1:12 (or $\dfrac{1}{12}$) as "12 of horizontal run for 1 of height" Knowing that dividing by $\dfrac{1}{12}$ is the same as multiplying by 12 (multiplying by the reciprocal)
Converting units of length (Grades 4–5)	Knowing that 1 ft = 12 in and 1 yd = 3 ft, and being able to put the rise and the lengths in the same unit
Square roots (Grade 8)	Knowing what a square root is and finding simple ones, such as $\sqrt{10000} = 100$
Right triangles and the Pythagorean theorem (Grade 8)	Knowing that in a right triangle $\text{base}^{2} + \text{height}^{2} = \text{hypotenuse}^{2}$ (example: 3, 4, 5) Picturing a right triangle with the horizontal run and the rise as two sides, where the hypotenuse is the length of the ramp board
Tangent and inverse tangent (high school Geometry and Precalculus)	Knowing that $\tan\theta$ is "height ÷ base" in a right triangle, which is exactly the slope Knowing that $\arctan$ ($\tan^{-1}$) returns the angle from a value of $\tan$ (a calculator or Excel can do the arithmetic)

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 horizontal run needed

Rise h (in)	6
Slope ratio n (slope 1:n)	12
Horizontal run needed L (in)	=B1*B2

Table to find the length along the slope

Rise h (in)	6
Horizontal run L (in)	72
Length along the slope s (in)	=SQRT(B2^2+B1^2)

Table to find the actual slope from the length you have

Rise h (in)	12
Horizontal run L (in)	120
Slope ratio n (slope 1:n)	=B2/B1
Percent slope p (%)	=B1/B2*100
Angle θ (°)	=DEGREES(ATAN(B1/B2))

Table to find the total length with landings

Horizontal run needed L (in)	432
Length of one landing d (in)	60
Number of landings m	1
Total length T (in)	=B1+B2*B3

Table to find the area and estimated cost

Length along the slope s (ft)	6.021
Ramp width w (ft)	3
Area A (ft²)	=B1*B2
Unit price u ($ per ft)	150
Estimated cost C ($)	=B1*B4

After pasting, the upper rows of column B are your inputs and the last rows (the bottom three rows in the third table, and B3 and B5 in the fifth table) are calculated automatically.
B3 in the first table shows 72, B3 in the second about 72.25, B3 to B5 in the third 10, 10 and about 5.71, B4 in the fourth 492, and B3 and B5 in the fifth about 18.06 and 903.15.
"SQRT" is the square root and "^2" squares a number. "ATAN" returns the arctangent, and because its answer is in radians, "DEGREES" turns it into degrees.
To use a price "per ft²" in the fifth table, change the formula in B5 to "=B3*B4".

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 horizontal run needed

Rise h (in)	6
Slope ratio n (slope 1:n)	12
Horizontal run needed L (in)	=B1*B2

Table to find the length along the slope

Rise h (in)	6
Horizontal run L (in)	72
Length along the slope s (in)	=SQRT(B2^2+B1^2)

Table to find the actual slope from the length you have

Rise h (in)	12
Horizontal run L (in)	120
Slope ratio n (slope 1:n)	=B2/B1
Percent slope p (%)	=B1/B2*100
Angle θ (°)	=DEGREES(ATAN(B1/B2))

Table to find the total length with landings

Horizontal run needed L (in)	432
Length of one landing d (in)	60
Number of landings m	1
Total length T (in)	=B1+B2*B3

Table to find the area and estimated cost

Length along the slope s (ft)	6.021
Ramp width w (ft)	3
Area A (ft²)	=B1*B2
Unit price u ($ per ft)	150
Estimated cost C ($)	=B1*B4

The same formulas as in Excel work as is (SQRT, ATAN and DEGREES have the same names). Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own.

How to calculate it in Python

import math

rise = 6           # rise h (in)
denominator = 12   # slope ratio n (slope 1:n)

run = rise * denominator                          # horizontal run needed L (in)
slope = math.sqrt(run ** 2 + rise ** 2)           # length along the slope s (in)
percent = rise / run * 100                        # percent slope (%)
angle_deg = math.degrees(math.atan(rise / run))   # angle (degrees). atan returns radians, so convert to degrees

print(f"Horizontal run needed: {run:.1f} in")
print(f"Length along the slope: {slope:.1f} in")
print(f"Slope: 1:{denominator} = {percent:.2f} % = {angle_deg:.2f} °")

# Going the other way: find the actual slope from the horizontal run you have, and compare it with 1:12
rise2 = 12            # rise (in)
available_run = 120   # horizontal run you have (in)
actual_n = available_run / rise2
target_n = 12
needed_run = rise2 * target_n
print(f"Actual slope: 1:{actual_n:.1f} (1:{target_n} needs {needed_run:.1f} in, difference {available_run - needed_run:.1f} in; negative means short)")

# Total length with landings
landing_len = 60     # length of one landing (in)
landing_count = 1    # number of landings
total = 432 + landing_len * landing_count
print(f"Total length with landings: {total:.1f} in")

Runs with the standard library only. math.sqrt() is the square root and math.atan() is the arctangent. Its answer is in radians, so math.degrees() turns it into degrees. Replace the rise and the slope ratio at the top with your own numbers and run it.

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

Horizontal run needed (rise × ratio n)

L = h × n

L = h\,n

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>L</mi>
    <mo>=</mo>
    <mi>h</mi>
    <mo>&#x2062;</mo>
    <mi>n</mi>
  </mrow>
</math>

L = h n

h*n

L := h*n;

L = h*n;

L = hn

Length along the slope (length of the ramp itself)

s = √(L² + h²),  L = √(s² − h²)

s = \sqrt{L^{2} + h^{2}},\quad L = \sqrt{s^{2} - h^{2}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>s</mi><mo>=</mo>
    <msqrt><msup><mi>L</mi><mn>2</mn></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup></msqrt>
    <mo>,</mo>
    <mi>L</mi><mo>=</mo>
    <msqrt><msup><mi>s</mi><mn>2</mn></msup><mo>-</mo><msup><mi>h</mi><mn>2</mn></msup></msqrt>
  </mrow>
</math>

s = sqrt(L^2 + h^2),  L = sqrt(s^2 - h^2)

{Sqrt[L^2 + h^2], Sqrt[s^2 - h^2]}

s := sqrt(L^2 + h^2);  L := sqrt(s^2 - h^2);

s = sqrt(L^2 + h^2); L = sqrt(s^2 - h^2);

s = √(L^2 + h^2), L = √(s^2 − h^2)

Actual slope from the length you have

n = L ÷ h,  p = h ÷ L × 100,  θ = arctan(h ÷ L)

n = \frac{L}{h},\quad p = \frac{h}{L} \times 100,\quad \theta = \arctan\frac{h}{L}

<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>θ</mi><mo>=</mo><mi>arctan</mi><mo>&#x2061;</mo><mfrac><mi>h</mi><mi>L</mi></mfrac>
  </mrow>
</math>

n = L / h,  p = h / L * 100,  theta = arctan(h / L)

{L/h, h/L*100, ArcTan[h/L]*180/Pi}

n := L/h;  p := h/L*100;  theta := arctan(h/L)*180/Pi;

n = L/h; p = h/L*100; theta = atand(h/L);

n = L/h, p = h/L × 100, θ = tan^(-1)(h/L)

Total length with landings

T = L + d × m

T = L + d\,m

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>T</mi><mo>=</mo><mi>L</mi><mo>+</mo><mi>d</mi><mo>&#x2062;</mo><mi>m</mi>
  </mrow>
</math>

T = L + d m

L + d*m

T := L + d*m;

T = L + d*m;

T = L + dm

Area and estimated cost (optional)

A = s × w,  C = s × u (or A × u)

A = s\,w,\quad C = s\,u\ (\text{or}\ A\,u)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>A</mi><mo>=</mo><mi>s</mi><mo>&#x2062;</mo><mi>w</mi>
    <mo>,</mo>
    <mi>C</mi><mo>=</mo><mi>s</mi><mo>&#x2062;</mo><mi>u</mi>
  </mrow>
</math>

A = s w,  C = s u

{s*w, s*u}

A := s*w;  C := s*u;

A = s*w; C = s*u;

A = sw, C = su

How to have ChatGPT do the calculation

You are an assistant for measurements in accessibility remodeling and outdoor construction. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

I am building a ramp over a 6 in rise. The slope will be 1:12.
Find each of the following:
1. The horizontal run needed (rise × slope ratio)
2. The length along the slope (Pythagorean theorem)
3. This slope as a percent and as an angle (use arctan and convert radians to degrees)
4. If I only have 60 in of horizontal run, the actual slope (the n in 1:n) and how much shorter that is than the run needed for 1:12

Show the formulas you used and the numbers from the execution result.

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Enter your numbers

Type the numbers you want to calculate with into the input fields
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\(h\)	aitch	The rise (the difference in height), from the first letter of "height". It is the height the ramp has to climb, such as from the ground to a porch or over a door threshold.
\(n\)	en	The slope ratio: the \(n\) when the slope is written 1:\(n\). It is a slope that rises 1 for every \(n\) of horizontal run, and \(n = L \div h\) (\(n = 12\) for 1:12). The larger \(n\) is, the gentler the slope.
\(L\)	el	The horizontal run. This page uses the first letter of "length". It is the length of the ramp seen from above, from the edge of the step to the end of the ramp, and \(L = h \times n\).
\(s\)	ess	The length along the slope (the true length). This page uses the first letter of "slope". It is the length of the ramp board or surface itself, found with \(s = \sqrt{L^{2} + h^{2}}\).
\(i\)	eye	The slope as a number (rise ÷ horizontal run), from the first letter of "inclination". \(i = 1/n\), and it has no unit.
\(p\)	pee	The percent slope, from the first letter of "percent". \(p = h \div L \times 100\), which is how much the ramp rises for every 100 of horizontal run (1:12 is about 8.33%).
\(\theta\)	theta	The angle of the ramp from the horizontal. Theta is the Greek letter often used for angles. It is found with \(\theta = \arctan(h \div L)\) (1:12 is about 4.76°).
\(T\)	tee	The total length with landings, from the first letter of "total". \(T = L + d \times m\), the horizontal length the whole ramp takes up.
\(d\)	dee	The length of one landing. This page uses the first letter of "depth" (how deep the landing is).
\(m\)	em	The number of landings. This page uses it as a count (it is not the unit meter, m).
\(w\)	double-u	The ramp width, from the first letter of "width". It is used for the area \(A = s \times w\).
\(A\)	ay	The ramp area, from the first letter of "area". It is the length along the slope × the width.
\(C\)	see	The estimated cost, from the first letter of "cost". It is the length along the slope (or the area) × the unit price.
\(u\)	you	The unit price, from the first letter of "unit price". Enter the price per ft, or per ft².
\(\tan\)	tangent	Tangent. In a right triangle it is height ÷ base. The slope \(i\) is exactly \(\tan\theta\).
\(\arctan\)	arctangent, inverse tangent (\(\tan^{-1}\))	The inverse of tangent. It returns the angle whose \(\tan\) is a given value, and is also written \(\tan^{-1}\). It is used to find the angle from the slope.
\(\sqrt{\ }\)	square root	Square root: the positive number that gives this number when squared. It is used with the Pythagorean theorem to find the length along the slope (example: \(\sqrt{25} = 5\)).
\(\approx\)	approximately equal to	The sign for "approximately equal". It is used when a value that does not come out evenly, such as a square root or an \(\arctan\), is rounded to a decimal.

Ramp Length Calculator (Rise and Slope, ADA 1:12)

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How to have ChatGPT do the calculation

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