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Roof Calculations of Slope, Rise, Run, AreaHow are roof rise, run, area or slope calculated?

• ROOF SLOPE CALCULATIONS - CONTENTS: how to calculate roof slope, rise/run, degrees, or tangents; how to calculate roof height over an attic floor at different places under a sloping roof; how to convert grade angle to percent slope; how to use tangents and inverse tangents with slopes.
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Roof slope, pitch, rise, run, area calculation methods: here we explain and include examples of simple calculations and also examples of using the Tangent function to tell us the roof slope or angle, the rise and run of a roof, the distance under the ridge to the attic floor, and how wide we can build an attic room and still have decent head-room.

This article series gives clear examples just about every possible way to figure out any or all roof dimensions and measurements expressing the roof area, width, length, slope, rise, run, and unit rise in inches per foot.

How to Calculate the Roof Slope (or any slope) Expressed as Rise & Run from Slope Measured in Degrees: fun with tangents

Question: if a roof slope is 38 degrees: what is the rise per foot or 12-inches of horizontal distance or "run"?

Complete details about converting slope or angle to roof, road, walk or stair rise & run along with other neat framing and building tricks using triangles and geometry are found
at FRAMING TRIANGLES & CALCULATIONS. And for a special use of right triangles to square up building framing, also see The 6-8-10 RULE

[Click to enlarge any image]

Reply: simple tricks with tangents get the roof, stair, road, or walk built to the specified slope

We can quickly convert any slope measured in degrees (or angle) using the basics of plane geometry. Don't panic. It's not really that bad if we just accept that basic plane geometry defines the relationships between a right triangle (that means one angle of the triangle is set at 90 degrees) and the lengths of its sides. a2 = b2 + c2 - the square of the length of the hypotenuse (a) equals the squares of the lengths of the opposite sides of a right triangle (b) and (c). Mrs. Revere, my elementary school teacher would be laughing if she were still alive.

Anyhow the magical trigonometry functions of tangent, cotangent, arctangent, sine, cosine, follow from basic geometry. Note: when using a scientific calculator to obtain a tangent value, enter the angle in degrees as a whole number such as 38, not 0 .38 or some other fool thing.

The TAN function can be used to convert a road grade or roof slope expressed in angular degrees to rise if we know the run, or run if we know the rise ONLY because we are working in the special case of a right triangle - that is, one of the angles of the triangle must be 90deg.

The trick for converting a slope expressed as an angle is to find the tangent of that angle. That number, a constant, lets us calculate rise if given run (say using a foot of run) or run if given the rise amount. That is, the Tangent of any angle is defined as the vertical rise divided by the horizontal run.

Tan <A = (Rise Y1) / Run (X)

Our sketch above shows how we calculate the roof rise per horizontal foot (12 inches) of run when we are given the roof slope in degrees (or as the roof pitch or angle expressed in degrees).The purple sloped line is the sloping roof surface. My vertical red lines show the rise (Y1) for each horizontal distance of one foot or 12" (not drawn to scale). It was trivial - I skipped digging into geometric calculations. I just took the given roof slope of 38 degrees and used my calculator (or a table, or actual geometry) to look up the value of Tan < A.

Tan 38o = 0.7813

Now using the formula above

0.7813= (Rise Y1) / Run (X)

We just rearrange the equation following the rules of algebra to find Rise V.

0.7813 x Run (X) = (Rise Y1)

We could now calculate any total rise we want. I'm calculating the rise per 12" of run:

0.7813 x 12" = 9.4"  rise per foot of run

The calculations in this show the total rise in inches (Y1) for every X1 or foot or 12" of horizontal run will be about 9.4" (actually 9.3756"). Hell we could calculate the total rise in the roof over say half the total width of the attic - that is the distance from the eaves to just under the ridge - that would tell us if I can stand up in the center of the attic of a roof with a 38 degree slope - for a given building width.

Checking the Tangent of a 12 in 12 slope: Tan 45o

As a sanity check we confirm that the tangent of 45 degrees is 1, or that two opposed 45 degree or 12 in 12 slope roof surfaces will form a 90 degree angle where they meet at the ridge, and will form 45 degree angles where they meet the wall top plate (or with respect to any horizontal line in the building).

Tan 45o = 1.00

Which is the same as saying a 45 degree slope = a 12 in 12 slope, or the roof will rise 12" for every 12" of horizontal run. We used this detail to calibrate our folding carpenter's rule scale for reading roof slope from the ground. Details of that procedure are
at ROOF MEASUREMENTS.

How to Calculate Roof Height Over an Attic Floor From Roof Slope & Building Width

[Click to see an enlarged, detailed version of any image

In geometry we learned that if we know the lengths of sides of a triangle, we can calculate its angles

. If we know two of its angles we can calculate the lengths of its sides.

And for a right triangle, the Tangent function gives some easy calculations of an unknown rise or run if I know the other two figures - the angle and either rise or run distance.]

The slope of our example roof is given as 38 degrees. And we figure that in calculating (or measuring) the "rise" of this same roof we can assume we are not so stupid as to not hold our tape vertical between the attic floor and the center of the ridge - so we can assume the other known angle is 90 degrees - we've got a nice "right triangle".

If my building width = 30 feet (chosen just for example) how much space do I have overhead in the center of the attic? Since our ridge is over the center of the attic that's the high point.

(Total building width / 2) = (30ft / 2) = 15 ft. total run or total horizontal distance from eaves to attic center under the ridge.

0.7813 x 15 ft = 11.7 ft total  rise across fifteen feet to the highest point in the attic.

Even if I'm Wilt the Stilt Chamberlin I can stand up in the center of this attic. I'm just six feet tall. Never mind Wilt, how far can I walk towards the eaves before I whack my head? We re-use the formula 0.7813= (Rise Y1) / Run (X) as follows?

0.7813 = (6 ft) / X where X is the run distance from the eaves where I will whack my bean. Rearranging using rules of algebra:

0.7813 x X = 6 ft

X = 6 ft. / 0.7813 = 7.6 ft.

At 7.6 ft. (that's about 7 ft. 7 in. when we convert decimal feet to inches) I can walk 7 ft. 7 in. from under the ridge before I need a band-aid. Doubling that I know we can build a room that is 14ft. 14in. or better, 15 ft. 2 in. wide and still have six feet of head-room. Neat, right?

How to Use Trivial Arithmetic to Convert Grade to Angle or Percent Slope

Grade, a figure used in road building, is simply slope or angle expressed as a percentage.

Rise / Run x 100 = Slope in Percent

Example:

If I build a sidewalk up the slope of a hill, the building department wants to know if I should have built stairs instead. If the slope, expressed in percent or percent grade is too steep, walkers are likely to slip, fall, and end this discussion. Suppose my sidewalk is 100 feet long and that the total rise from the low end to the high end of the walk is four feet:

4 ft. / 100 ft. x 100 = 4% Grade - which my inspector accepted as ok. Typical building codes specify that

For pedestrian facilities on public access routes, the running grade of sidewalks will be a maximum of 5%.

By "running grade" we mean that at no point in the sidewalk will the grade be steeper than 5%. In case it's not obvious, that means we'd see a 5 foot rise in100 feet of horizontal travel if the walk were sloped uniformly over its entire length.

Definition & Uses of Tangent & Tan-1 when Working With a Right Triangle (building roofs, stairs, walks, or whatever)

A tangent is the ratio of two sides of a right triangle: specifically the height (Y) divided by the base or length (X). For any given stair slope (or angle) or triangle slope (angle T or "Theta" as we say in geometry class), that ratio remains unchanged.

Or in geometry speak:

Height Y1 / Length X1 = Height Y2 / Length X2

as long as we keep the slope or angle unchanged.

The tangent function is a ratio of horizontal run X and vertical rise Y. For any stairway of a given angle or slope (say 38 degrees in your case) the ratio of run (x) to rise (y) will remain the same.

That's why once you set your stair slope (too steeply) at 38 degrees, we can calculate the rise or run for any stair tread dimension (tread depth or run or tread height or riser) given the other dimension (tread height or rise or tread depth or run).

The magic of using the Tangent function is that we can use that ratio to convert stair slope or angle in degrees to a number that lets us calculate the rise and depth or run of individual stair treads

• In roof speak we describe this slope or ratio as roof slope (Rise / Run).
• In stair speak we describe this ratio as (stair riser height / stair tread depth) or as (stairway total rise / stairway total run).
• In sidewalk and road building speak we describe this ratio as the grade or percentage of slope (which is TAN x 100).

Here are two examples of roof pitch expressed as horizontal run and riser vertical change in height (rise) for a roof with with a 38 degree slope: :

• On a 38 degree sloping roof (angle T) each individual vertical rise of 9.4" (Y1) would have a horizontal run (X1) of 12 inches.
• Total roof rise or change in elevation for a 38 degree sloping roof (angle T) with one "giant" rise or step of 7.8 feet to the center of the attic (Y2) would have total a run (X2) of 10 feet. (In these calculations, as long as we keep the same unit for both rise and run we can change among inches, feet, meters, or roofing hammer handle lengths - whatever.

The magic is that the tangent ratio of the rise over run (Y/X) for roofs with different run lengths would always be the same - because they are built to the same slope or angle. You can see that reflected in our drawings above.

For a special use of right triangles to square up building framing, also see The 6-8-10 RULE - a simple method for assuring that framing members have been set at right angles to one another.

How to Calculate the Tangent Value rather than Looking it Up

Could we calculate the tangent of 38 degrees? Well it's easier to use a scientific calculator and just ask for the Tangent of a known angle.

If we knew that we had a triangle of 38 degrees at angle T (Theta) and if we knew two specific measurements X and Y we could indeed calculate T = Y/X. After all, the tangent of angle Theta is the ratio of Y/X.

I used an online calculator available at http://www.creativearts.com/scientific calculator/ and the simple formula shown in my illustration.

I got also some help (a refresher on geometry) from Ferris High school's excellent geometry department who provides a more detailed analysis of the same problem as that posed by George Tubb's question.[17]

Use Inverse Tangent, Tan-1, Arctan or Arctangent function to compute slope or angle from rise and run of a roof or other slope.

Those Ferris High kids in Spokane can also show you how to work this problem in the other direction: that is, if we know the rise and run of the roof we can calculate its slope or angle in degrees by using the arctangent function. Purists and mathematicians argue that the inverse tangent function (Tan-1) commonly found on calculators and used to convert a Tangent value back into degrees of slope is not identical to the true definition of Arctangent.

In several of our roofing and stair building measurement & calculation articles including

ARCTANGENT CALCULATES ROOF / STAIR ANGLE where we usethe Inverse Tangent, Tan-1, Arctan or Arctangent functioncalculate the angle of a set of stairs when we know the stair rise and run

and

FRAMING TRIANGLES & CALCULATIONS

and

FROGS HEAD SLOPE MEASUREMENT we demonstrate the use of both TAN and (TAN-1) .

Table of Roof Slopes, Roof Types, Walkability

This table has moved to a new page at ROOF SLOPE TABLE, TYPES, WALKABILITY

...

Continue reading at ROOF SLOPE DEFINITIONS or select a topic from closely-related articles below, or see our complete INDEX to RELATED ARTICLES below.

Or see ROOF MEASUREMENTS where we describe all of the methods for measuring roof slope, area, etc.

Or see these

Building & Roof Measurement Articles

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