Roof Calculations of Slope, Rise, Run, Area
How 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 Use the 6-8-10 Rule
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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. a^{2} = b^{2} + c^{2} - 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 Y_{1}) / 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 (Y_{1}) 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 38^{o} = 0.7813
Now using the formula above
0.7813= (Rise Y_{1}) / Run (X)
We just rearrange the equation following the rules of algebra to find Rise V.
0.7813 x Run (X) = (Rise Y_{1})
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 (Y_{1}) 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 45^{o}
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 fdorm 45 degree angles where they meet the wall top plate (or with respect to any horizontal line in the building).
Tan 45^{o} = 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
In geometry, 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.
[Click to see an enlarged, detailed version of any image]
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 tota 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 Y_{1}) / 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. whenwe 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.
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" (Y_{1}) would have a horizontal run (X_{1}) 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 (Y_{2}) would have total a run (X_{2}) 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 Use 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/scientificcalculator/ 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 and also at FROGS HEAD SLOPE MEASUREMENT we demonstrate the use of both TAN and (TAN^{-1}) .
More Reading
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Books & Articles on Building & Environmental Inspection, Testing, Diagnosis, & Repair
"Choosing Roofing," Jefferson Kolle, January 1995, No. 92, Fine Homebuilding, Taunton Press, 63 S. Main St., PO Box 5506, Newton CT 06470 - 800-888-8286 - see http://www.taunton.com/FineHomebuilding/ for the magazine's website and for subscription information.
Owens Corning Corporation, One Owens Corning Parkway
Toledo, Ohio 43659
U.S.A.
Telephone: (419) 248-8000
Fax: (419) 248-5337
http://www.owenscorning.com Owens Corning is credited as the inventor of fiberglass when Owens Illinois [O-I] researcher Dale Kleist and his colleague John Thomas stumbled onto and then realized the significance of producing glass fibers in 1932. O-I formed a joint venture with the Corning Glass Works in 1935, leading to the formation of Owens Corning Corporation in 1938. More on Owens Corning's history is at
Focus, Toledo, Ohio, Owens-Corning Fiberglas Corporation, October 1988.
"A History of Innovation," http://www.owenscorning.com, 1997.
Stewart, Thomas A., "Owens-Corning: Back from the Dead," Fortune, May 26, 1997.
International Directory of Company Histories, Vol. 20. St. James Press, 1998.
"Two-Year Wisconsin Thermal Loads for Roof Assemblies and Wood, Wood–Plastic Composite, and Fiberglass Shingles [on file as Roof_Thermal_Loads.pdf] - ",
Jerrold E. Winandy
Michael Grambsch
Cherilyn A. Hatfield, US Department of Agriculture, US Forest Products Laboratory, Research Note FPL-RN-0301
ARMA - Asphalt Roofing Manufacturer's Association - http://www.asphaltroofing.org/
750 National Press Building, 529 14th Street, NW, Washington, DC 20045, Tel: 202 / 207-0917
ASTM - ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA, 19428-2959 USA The ASTM standards listed below can be purchased in fulltext directly from http://www.astm.org/
NRCA - National Roofing Contractors Association - http://www.nrca.net/, 10255 W. Higgins Road, Suite 600,
Rosemont, IL 60018-5607, Tel: (847) 299-9070 Fax: (847) 299-1183
UL - Underwriters Laboratories - http://www.ul.com/
2600 N.W. Lake Rd.
Camas, WA 98607-8542
Tel: 1.877.854.3577 / Fax: 1.360.817.6278
E-mail: cec.us@us.ul.com
copy on file as /roof/Roofing_Historic_NPS .pdf Roofing for Historic buildings", Sarah M. Sweetser, Preservation Brief 4, Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source:
http://www.nps.gov/history/hps/tps/briefs/brief04.htm
copy on file as /roof/Asbestos-to-Zinc_Metal_Roofing_NPS .pdf From Asbestos to Zinc, Roofing for Historic buildings, Metals", Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source:
http://www.nps.gov/history/hps/tps/roofingexhibit/metals2.htm
copy on file as /roof/Asbestos-to-Zinc_Metal_Roofing_NPS_3 .pdf From Asbestos to Zinc, Roofing for Historic buildings, Metals-part II, Coated Ferrous Metals: Iron, Lead, Zinc, Tin, Terne, Galvanized, Enameled Roofs", Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source:
http://www.nps.gov/history/hps/tps/roofingexhibit/metals.htm
copy on file as /roof/Asbestos-to-Zinc_Metal_Roofing_NPS_4 .pdf From Asbestos to Zinc, Roofing for Historic buildings, Metals-part III, Slate", Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source:
http://www.nps.gov/history/hps/tps/roofingexhibit/slate.htm
copy on file as /roof/Asbestos-to-Zinc_Metal_Roofing_NPS_5 .pdf From Asbestos to Zinc, Roofing for Historic buildings, Metals-part IV, Wood", Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source:
http://www.nps.gov/history/hps/tps/roofingexhibit/slate.htm
copy on file as /roof/Asbestos-to-Zinc_Metal_Roofing_NPS_5 .pdf From Asbestos to Zinc, Gutters", Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source:
http://www.nps.gov/history/hps/tps/roofingexhibit/gutters.htm
copy on file as /roof/Asbestos-to-Zinc_Metal_Roofing_NPS_2 .pdf From Asbestos to Zinc, Roofing for Historic buildings, Metals- Roofing Today", Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source:
http://www.nps.gov/history/hps/tps/roofingexhibit/roofingtoday.htm
/exterior/NPS_Preserv_Brief_16_Subs_Mtls.pdf The Use of Substitute Materials on Historic Building Exteriors ",
Sharon C. Park, AIA, Preservation Brief 16, Technical Preservation Services, National Park Service, U.S. Department of the Interior, web search 9./29.10, original source: http://www.nps.gov/history/hps/tps/briefs/brief16.htm
Books & Articles on Building & Environmental Inspection, Testing, Diagnosis, & Repair
Our recommended books about building & mechanical systems design, inspection, problem diagnosis, and repair, and about indoor environment and IAQ testing, diagnosis, and cleanup are at the InspectAPedia Bookstore. Also see our Book Reviews - InspectAPedia.
The Home Reference Book - the Encyclopedia of Homes, Carson Dunlop & Associates, Toronto, Ontario, 25th Ed., 2012, is a bound volume of more than 450 illustrated pages that assist home inspectors and home owners in the inspection and detection of problems on buildings. The text is intended as a reference guide to help building owners operate and maintain their home effectively. Field inspection worksheets are included at the back of the volume. Special Offer: For a 10% discount on any number of copies of the Home Reference Book purchased as a single order. Enter INSPECTAHRB in the order payment page "Promo/Redemption" space. InspectAPedia.com editor Daniel Friedman is a contributing author.
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Architectural elements: the technological revolution: Galvanized iron roof plates and corrugated sheets; cast iron facades, columns, door and window caps, ... (American historical catalog collection), Diana S Waite, available used out of Amazon.
Green Roof Plants: A Resource and Planting Guide, Edmund C. Snodgrass, Lucie L. Snodgrass, Timber Press, Incorporated, 2006, ISBN-10: 0881927872, ISBN-13: 978-0881927870. The text covers moisture needs, heat tolerance, hardiness, bloom color, foliage characteristics, and height of 350 species and cultivars.
Green Roof Construction and Maintenance, Kelley Luckett, McGraw-Hill Professional, 2009, ISBN-10: 007160880X, ISBN-13: 978-0071608800, quoting: Key questions to ask at each stage of the green building process Tested tips and techniques for successful structural design
Construction methods for new and existing buildings
Information on insulation, drainage, detailing, irrigation, and plant selection
Details on optimal soil formulation
Illustrations featuring various stages of construction
Best practices for green roof maintenance
A survey of environmental benefits, including evapo-transpiration, storm-water management, habitat restoration, and improvement of air quality
Tips on the LEED design and certification process
Considerations for assessing return on investment
Color photographs of successfully installed green roofs
Useful checklists, tables, and charts
Problems in Roofing Design, B. Harrison McCampbell, Butterworth Heineman, 1991 ISBN 0-7506-9162-X (available used)
Roofing The Right Way, Steven Bolt, McGraw-Hill Professional; 3rd Ed (1996), ISBN-10: 0070066507, ISBN-13: 978-0070066502
Slate Roofs, National Slate Association, 1926, reprinted 1977
by Vermont Structural Slate Co., Inc., Fair Haven, VT 05743, 802-265-4933/34. (We recommend this book if you can find it. It
has gone in and out of print on occasion.)
Roof Tiling & Slating, a Practical Guide, Kevin Taylor, Crowood Press (2008), ISBN 978-1847970237, If you have never fixed a roof tile or slate before but have wondered how to go about repairing or replacing them, then this is the book for you. Many of the technical books about roof tiling and slating are rather vague and conveniently ignore some of the trickier problems and how they can be resolved. In Roof Tiling and Slating, the author rejects this cautious approach. Kevin Taylor uses both his extensive knowledge of the trade and his ability to explain the subject in easily understandable terms, to demonstrate how to carry out the work safely to a high standard, using tried and tested methods.
This clay roof tile guide considers the various types of tiles, slates, and roofing materials on the market as well as their uses, how to estimate the required quantities, and where to buy them. It also discusses how to check and assess a roof and how to identify and rectify problems; describes how to efficiently "set out" roofs from small, simple jobs to larger and more complicated projects, thus making the work quicker, simpler, and neater; examines the correct and the incorrect ways of installing background materials such as underlay, battens, and valley liners; explains how to install interlocking tiles, plain tiles, and artificial and natural slates; covers both modern and traditional methods and skills, including cutting materials by hand without the assistance of power tools; and provides invaluable guidance on repairs and maintenance issues, and highlights common mistakes and how they can be avoided.
The author, Kevin Taylor, works for the National Federation of Roofing Contractors as a technical manager presenting technical advice and providing education and training for young roofers.
The Slate Roof Bible, Joseph Jenkins, www.jenkinsslate.com,
143 Forest Lane, PO Box 607, Grove City, PA 16127 - 866-641-7141 (We recommend this book).
Solar heating, radiative cooling and thermal movement: Their effects on built-up roofing (United States. National Bureau of Standards. Technical note), William C Cullen, Superintendent of Documents, U.S. Govt. Print. Off (1963), ASIN: B0007FTV2Q
"Weather-Resistive Barriers [copy on file as /interiors/Weather_Resistant_Barriers_DOE.pdf ] - ", how to select and install housewrap and other types of weather resistive barriers, U.S. DOE