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Calculating Guide for Roofing Sq. Footage Substitute the formulas given under each type of roof to obtain actual square foot area for the following roofs. Formula refers to shaded area in diagram. Add 10% approximate extra for waste. 1.Multiply length x width of building; include overhangs. Multiply this area x slope factor. Calculate the Surface Area of the Roof. You’ll notice as an area is drawn the area and perimeter results are displayed in the bottom right corner. With a single pitch roof you’ll have only one area result that relates to the entire footprint or perimeter of the roof.
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Roof area calculation & measurement methods:
Here we describe various methods for measuring all roof data: roof slope or pitch, rise, run, area, and other features. We include on-roof measurements, roof measurements or estimates that can be made from ground level, and several neat tricks using a folding ruler to measure roof angle or slope.
This article shows how simple measurements can give the roof area without having to walk on the roof surface. 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.
We also provide a MASTER INDEX to this topic, or you can try the page top or bottom SEARCH BOX as a quick way to find information you need.
How to Calculate the Area of a Roof
The attractive New Zealand slate roof shown here protects the beloved Catholic Church of St Werenfried in Waihi Village, at the edge of southern Lake Tekapo, in New Zealand (South Island).
Making one or two simple straightline measurements from the ground along with clever use of a folding rule or other methods discussed in this article series can give us accurate measurements of the church roof dimensions, slope, and area.
[Click to enlarge any image]
Definition of Roof Dimensions
Roof rise (b): first we will obtain the total roof rise by counting siding courses. We measure siding width & then we count courses at the building gable end. If we start counting siding at a horizontal lilne even with the lower roof edge or eaves and count up to the ridge, we've got a close guess at the total roof rise.
The number of siding courses from the roof triangle base to the roof peak x siding course width = total roof rise = (b)
Roof length (c): Measure or step off building gable end width.
Roof width (a): This data allows us to calculate the roof triangle as we know two sides (b) and (c) of the three sides of a right triangle (the red lines in our photo at above left). Let (b) = the vertical rise in the roof and (c) = the roof length (building length + gable overhangs). The third side of the triangle, its hypotenuse or the sloping surface of the roof, or side is (a) which is calculated as follows:
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).
Given a2 we use our calculator to take the square root and bingo, we have the length of the sloping side of the roof.
Given that we now know all of the lengths of our triangle we can easily obtain roof slope too if we need it.
Now finally to get the roof area, we just need one more figure, the length of the roof along the building eaves or ridge. From the ground we measure or step off building length (L).
The roof area(RA) is calculated easily:
We multiply the Roof Length (c) (which is the sum of building length plus the gable end overhangs of the roof) by the Roof Width of slope (a) that we just figured out above when we computed the hypotenuse of the roof triangle (that's why we needed the roof rise number).
RA = (a) x (c)
We can use the TAN or tangent feature of a calculator as a trivial way to convert degrees of slope (or grade if we're building a sidewalk or road) into units of run per unit of rise. The Tangent of any angle expressed in degrees is nothing more than a ratio:
Tangent = Rise / Run
How to calculate rise per foot of run for this roof using the number from an angle level
I'll show that even if we screw up we can still come out ok finding the angle and then the rise and run of a roof using the angle finding level.
I read 81 deg. on my angle level. Now let's figure run for 12' of rise for an 81 degree slope - HOLD ON! something's crazy here. This is a low slope roof, how can it be sloping 81 degrees? Egad! that's nearly straight up! This is a good lesson in thinking for yourself - or performing a sanity check on calculations.
The answer is I was holding my angle level on the wrong scale. I could have made my photos over again holding the angle level the right way, but there's an easier trick:
81 degrees is just 9 degrees off of dead vertical (90 - 81 = 9). So really I could go just 9 degrees off of flat. As 'flat' is 0 degrees of slope, flat+ 9 = 9. My roof actually slopes 9 degrees. Whew!
The Tan value for my 9 degree slope roof = Tan ( 9) = 0.1583
Find Tan 9 deg using a handy dandy calclator, or table such as the one I give
at ROOF MEASURE by FOLDING RULE.
The inches of rise for 12-inches of run on a 9 deg low-slope roof is calculated as follows:
Tangent is defined as a ratio: Rise / Run so all we need is a little algebra (don't faint, it's easy):
0.1583 = Rise / Run
Set run to 12-inches because we're going to calculate the rise per foot of run.
0.1583 = 12 / Run
Use simple algebra:
0.1583 x Run = 12' of rise
Run = 12' / 0.1583
Run = 75.8'
That makes sense: we travel about 75 inches horizontally for every 12 inches of vertical rise on this low slope 9 degree roof.
Calculate Unit Rise for the Roof Rise
To calculate total rise if I knew the total run (say we had made an on-roof measurement) we take the following steps:
Total Rise = (Total Run in Feet) x (Rise per Foot)
The Tan value for my 9 degree slope roof = Tan ( 9) = 0.1583
0.1583 = Rise / Run
Using a little high school algebra we can re-write the equation as
0.1583 x Run = Rise
If I want to know the rise per foot of run I calculate
0.1583 x 12 = 1.89 ' of rise per foot of run.
Calculate Total Rise for the Roof
I measured the total horizontal run - my building width is 20 ft. + a total of 2 ft. of overhang at the eaves.
0.1583 x 22 = 3.5 ft.
My roof increases in height 3.5 ft. from the eaves to the high end (this is a shed roof).
I can check this result against the rise per foot we got above.
(22 ft. x 1.89' rise per foot) / 12 = 3.5 ft. (thank goodness)
For the Tangential Enthusiast - Usnig Inverse Tangent Function Tan-1
The inverse Tan-1 function can convert a Tan value back into degrees of roof slope.
Tan-1 (1.43) = 55 deg. and wonderfully, Tan-1 (1.00) = 45 deg.
Since Tan is a simple ratio of unit Rise / unit Run, we note that we can quickly convert a roof slope in degrees into the number of inches of rise per 12' of run as follows, using a 55 deg. slope as example:
Tan (55) = 1.43
Since 1.43 = rise / run we can use simple algebra to write:
1.43 x 12' run = 17.16' of rise per 12' of run
Summary of How Roof Measurements are Calculated![]() How to Measure or Estimate the Total Roof Area
If you have safe access to the roof surface you can quickly make the needed area measurements: just measure from the ridge to the lower edge or eaves, keeping your tape straight.
With a decent 3/4' or 1' wide 30 ft. tape measure you can extend the tape out to catch the roof eaves without having to walk dangerously close to the roof edge. Also measure the roof edge or length.
Roofers measure or estimate the total roof area in square feet that is then converted to roofing squares - the unit of ordering of roofing material. One roofing 'square' covers 100 sq.ft. of roof area. Convert roof area in square feet to squares of roofing material by dividing by 100.
Watch out: do not walk on roofs that are fragile (you will damage the surface, make leaks, and make people mad.) Do not try to access a roof that is unsafe for any reason: height, slope, condition, wet, slippery, windy, etc. In those conditions you'll be better off making a few simple measurements from the ground level to figure the roof areas involved.
Estimating the Roof Area for Complex Roofs
Watch out: also that roof measurement is only trivial for simple shed or gable roofs whose slopes are a simple rectangle. For hipped roofs, mansards, and intersecting gables some simple triangles need to be measured if you want an accurate estimate of roof area.
Accurate Calculation of the Area of an Individual Roof Slope from Available Measurements
To be more accurate, and in cases where we need to get the roof area while working from the ground we can get the actual or accurate area of an individual roof slope as follows
How to Use Horizontal or 'Flat' Roof Projections as Rough Estimates of Roof Area for Inaccessible Roofs
Another simplistic approach used by some estimators is to ignore complex roof structure, just measuring the building's footprint and the roof slope - an approach that gets you into the right 'ballpark' but will very seriously under-estimage the roof area for steep slope roofs.
Frankly, as we illustrate beginning at ROOF MEASUREMENTS, there are some easy and accurate alternatives that can give a good estimate of roof area while making measurements only from the ground. But to understand how some people use a flat or horizontal projection of a roof to guess at roof area, here is the procedure.
BF: Measure the building footprint or BF
EO: Measure or estimate the increase in footprint size given by the roof eaves overhang. (Tip: look at the drip line under the roof eaves and measure the distance from the outer edge of the drip line to the building exterior wall. This is EF.
GO: Measure or estimage the increase in footprint size given by the gable end overhangs. This is GO.
RF: If the eaves overhang and gable end overhang are the same on both front and back and left and right building ends we just add these up to obtain Roof Footprint or RF.
RF = BF + (2 x EO) + (2 x GO)
RA: Obtain the approximate roof slope to convert Roof Footprint to Roof Area - RA using
the ROOF SLOPE MULTIPLIER TABLE given below.
RA = RF x Roof Slope Multiplier
All of the Ways to Get the Roof Slope
You can calculate the roof slope if you know just a few measurements. Details are
at ROOF SLOPE CALCULATIONS
You can estimate the roof slope from the ground by any of several methods described
at ROOF MEASUREMENTS How to convert the building footprint or roof 'footprint' (building footprint + roof overhangs) to roof area
To convert the rectangular footprint of the building roof to roof area we need to increase the footprint area to account for the greater area covered by the sloping roof. Using any of the roof slope estimating or measuring methods described above, just this simple roof slope multipication chart:
...
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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.
We also provide a MASTER INDEX to this topic, or you can try the page top or bottom SEARCH BOX as a quick way to find information you need.
How to Calculate the Roof Slope (or any slope) Expressed as Rise & Run from Slope Measured in Degrees: fun with tangentsArticle Series Contents
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 FRAMING SQUARE UP 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 smiling 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.
Definition of Tangent: the Tangent of any angle is defined as the vertical rise divided by the horizontal run.
A tangent is also defined as a ratio of two length: the length of the side of the triangle opposite the angle divided by the length of the adjacent side of the angle.
For a right triangle (such as a roofing triangle) in which the roof rise is the vertical line opposite the roof angle or slope, the tangent of the roof angle is the ratio of roof rise over the length of the sloped roof surface.
In the illustration above the tangent T of the 38 degree angle (the roof slope) is the ratio: roof rise (Y2) / Sloped Roof Run (X2)
So we write the tangent of angle A as:
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').
Heck 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 totalrise 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
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: :
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 FRAMING SQUARE UP 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 function to calculate the angle of a set of stairs when we know the stair rise and run
and
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 SLOPE CALCULATION FAQs - questions and answers posted originally on this page.
ROOF MEASUREMENTS where we describe all of the methods for measuring roof slope, area, etc.
Or see these
Building & Roof Measurement Articles
Suggested citation for this web page
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INDEX to RELATED ARTICLES: ARTICLE INDEX to BUILDING ROOFING
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