How To Find The Tangent Using A Calculator Or A Table
A tangent is also defined as a ratio of two lengths: the length of the side of the triangle opposite the angle divided by the length of the adjacent side of the angle.
Watch out: 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 or table can be used to convert a road grade or roof slope or stair 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.
You can also find free tables of every trigonometric function online, such as a tangent table that will give the value of the tangent for any angle.
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%.
Calculate The Square Footage Of A Gable
To calculate the square footage of a gable wall section, you need to know the wall length and the height of the gable. If for some reason you can’t get the height of the gable, you don’t have a ladder long enough, or you are one of those that should not be at heights if you know the pitch of the roof and the wall length, you can also calculate the square footage of a gable.
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How To Calculate Roof Area
To calculate your roof area, simply multiply your roof length by your roof slope height, and multiply this by two. This should give you the total area of your roof, not accounting for a chimney or other sections of your roof not covered by tiles
Things You Will Need

Square footage of your house

Pitch of your roof
Tip
This calculation will give a solid estimate for a single slope roof and a simple gable roof with a single pitch. For more complex roof lines with multiple pitch angles, its a little more complicated but the same steps can be followed just break down the roof into separate sections with a different square footage for each different pitch angle.
Find The Slope Of The Roof
Before you can determine the area of a roof, you need to know its slope. You can find the slope by getting up on a ladder at the rake edge of the roof and measuring with a ruler and a spirit level, or you can use a speciallydesigned slope finder.
Measuring roof slope with a bubble level and a ruler. This level has no measurement markings, so the yellow tape is being used to mark 12 inches from the end of the level that touches the roof. This roof has an 11in12 slope.
You could also take your measurement in the attic using the underside of a rafter, the underside of the roof sheathing, or even the ceiling, if youre sure that it follows the slope of the roof.
Eyeballing the slope and guessing is probably not a good idea.
The result you get from guessing like this might be close enough for government work, but if its a large roof and youre a roofer with a 10% profit margin, I wouldnt recommend it.
There are also a number of smartphone apps that you can use to find the slope of your roof, if you have a clear view of a gable end on the building. These can be acceptably accurate sometimes.
You dont have to get up on the roof or go to the attic to use these. Go search in your app store for roof pitch apps. Youll see.
Measuring roof slope with a slope finder.
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Estimating The Cost Of The Roof
Once you know the area of the roof, you have to decide on the scope of works. Do you want to build the roof from scratch, or maybe replace the old layers? You also need to decide on the finishing material. For example, asphalt shingles will be less expensive than clay tiles.
After you’ve made all of these decisions, it’s best to contact your local contractor and ask about the cost of both materials and labor per square meter or square foot of roof area. Remember that this unit price should include all additional elements such as fasteners, as well as the cost of removing all layers and disposing of them .
The most typical cost of roofing falls between $2 and $4 per square foot . If the price proposed by your contractor deviates from these values, it is advisable to check with other contractors before starting the works.
If you cannot contact any contractor, calculate the cost of materials and multiply this price by 2.5. You will obtain an approximate price including labor, assuming that 40% of the cost is taken by the materials and 60% by the labor.
The last thing you have to do is to multiply this cost by your roof area and voilà, you now know the total cost of your roof! Remember that the real cost might differ from this value a bit due to waste, so plan ahead and assign more funds to this construction work.
If you are not on a tight budget, maybe you’d like to add solar panels?
Also, make sure to take a look at our concrete slab cost calculator, too!
How Do You Calculate The Square Footage Of Your Roof
Is it time for a roof replacement? Are you trying to figure out how much youll have to invest in your new roof?
If so, youre probably overwhelmed trying to learn everything that goes into the cost of your roof replacement. To help you understand the size of your roof investment, you first need to know the size of your roof. Thats why were going to break down how to measure your roofs square footage.
The team at Bill Ragan Roofing has been repairing and replacing roofs in Nashville and surrounding areas since 1990. We take extreme pride in everything we do, especially our workmanship.
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How To Calculate The Roof Slope Expressed As Rise & Run From Slope Measured In Degrees: Fun With Tangents
Here we step through four methods for calculating vertical rise, horizontal run distance, or the slope of a roof or stairway or any other sloping construction measured in degrees or expressed more commonly as rise/run.
For example a 4/12 roof means that the roof height rises 4 inches for every 12 inches of horizontal distance or “run” from the roof edge or eaves towards the roof peak or ridge.
For a roof the total rise is the vertical distance under the ridge to the attic floor. This height tells us how wide we can build an attic room and still have decent headroom.
For a stairway the total rise is the vertical distance between the lower and upper walking surfaces joined by the stairs and the stairway run is the horizontal distance occupied by the stairway.
We will explain how to use the Tangent function to tell us the roof or stairway slope or angle, its vertical rise and horizontal run.
We show how to determine the height of a roof if we know its slope or angle.
We also translate lesswidelyused slope figures like percent grade to an angle or slope of a roof, stair, or a roadway.
Article Series Contents
Simple tricks with tangents get the roof, stair, road, or walk built to a specified slope
We can quickly convert any slope measured in degrees 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 and the lengths of its sides.
Squares Rectangles And Triangles
Nearly every roof, no matter how simple or complicated, can be broken down into squares, rectangles or triangles. You can draw lines on your diagram from the ridge to the eaves to make triangles out of the hips. This will leave you with four triangles for the hips, two rectangles for the rest of the hip roof, and another rectangle for the leanto roof.
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Calculating The Amount Of Material For A Gable Roof
As mentioned above, the area and height of the pediment is determined for the purchase of facing materials. According to the known parameters of this part of the structure, it is quite possible to find out how much brick or siding is required for facing. The exact area of finishing is equal to the difference between the total area of the pediment and the area of the window openings.
Measuring The Roofs Pitch
The incline or pitch of a roof can be easily measured with a level, tape measure, and a pencil. The roofs pitch is the number of inches the roof rises in 12 inches. Therefore, mark off 12 inches on the level and place it down horizontally against the roof rafter.
Next, measure vertically from the 12inch mark on the level straight up to the underside of the rafter, as illustrated below. That measurement is the number of inches the roof rises in 12 inches.
Some homes will have roofs with multiple pitches. Porches and dormer windows, for example, can have a different pitch from that of the rest of the house. For an accurate surface area measurement you should make note of these additional structures.
Using the table below, lookup the pitch/es and make a note of the number/s in the multiplier column. This multiplier number will be used to multiply the surface area calculation that SketchAndCalc will provide.
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The Need For Accurate Calculations
Improper sizing truss system threatens overruns or conversely a lack of materials
To begin construction, you need to know how much material is required to form the roofing pie and coatings. Given that the price of building materials is high, the calculations must be made more precise in order to avoid overspending. Calculate the amount of material with a minimum of errors as possible, knowing the area of the roof.
Notes On Calculating The Area Of A Roof
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Roof Area Calculation Guide
All you need to do is to follow some steps
 Divide your roof pitch by 12. It means roof pitch is the inches in a change of height over the distance of one foot. Dividing by 12 will give you the ratio of inches rising every distance. For instance, the standard roof pitch is about 4. Hence for the pitch angle of 4, divide it by 12
 Square your result of step 1, and so keeping with the roof pitch angle of 4, it gives 1/9.
 Add one and then square root it. Adding one to the 1/9 will give you 10/9, and the square root of 10/9 will give 1.0541.
 Multiply by the square footage of the single house floor. Let us continue the example if you have a twofloor house having a total square foot of 2700 square feet. It means each floor has 1350 square feet, so multiply 1350 by 1.0541 for getting the result of 1423 square feet. Thus, you will need 1423 square feet of roofing.
However, this is a simple concept that you have to keep in mind. Let us start briefing steps to calculate the area of the roof from scratch.
Whats Your Roof Slope
You will also need to know the slope of your deck.
To determine this, measure the vertical rise of your deck in inches over a 12 horizontal distance.
If this rise is 4, then your roof slope is 4 in 12.
Roof slopes are always expressed with the vertical rise mentioned first and the horizontal run mentioned second.
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Calculate The Complex Roof Areas
Most roofs in Canada would be considered on the complex side. Multiple adjacent pitches, dormers, and turrets are common. Keep in mind that an experienced metal roofing contractor will need to factor in extended labour time for areas of your roof that will require precision metalworking skills.
Start by finding the footprint area and multiply by the correct slope factor determined by pitch . In Figure 4 below, the dashed blue lines give a simplified view of the footprint. Multiply L x W to calculate the area for each rectangular section. Note the area of 77ft2 that needs to be subtracted for the recessed area.
Once the footprint including overhang areas have been determined for each area, find the appropriate slope or pitch factor in the chart below. For example, a roof area with an 8/12 pitch has a slope factor of 1.202 so 2053ft2 x 1.202 = 2468ft2 total roof area.
A Note about complex roofs: Complex roofs need more accessories and trims thaneasy roofs resulting in a slightly higher total cost/square foot.
Figure 4. Calculating Complex Roof Area
What Is The Average Quote For Getting A Roof Pitched
There are a lot of factors that will affect how much getting your roof pitched will cost design, style, accessibility, cost of labor etc. In the USA, you can expect to pay between $3.19$4.43 per sq.ft. for asphalt shingles. If you choose another material, such as metal or wood, the range is $7.35$10.73 per sq.ft.
In the UK, the price of getting a roof fitting is £72£75 per square meter. These are only estimates though, talk to your neighbors to see what roofing costs in your area.
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Definition & Uses Of Tangent & Tan
A tangent is the ratio of two sides of a right triangle: specifically the height divided by the base or length . For any given stair slope or triangle slope , 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 the ratio of run to rise will remain the same.
That’s why once you set your stair slope at 38 degrees, we can calculate the rise or run for any stair tread dimension given the other dimension .
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 .
 In stair speak we describe this ratio as or as .
 In sidewalk and road building speak we describe this ratio as the grade or percentage of slope .
Here are two examples of roof pitch expressed as horizontal run and riser vertical change in height for a roof with a 38 degree slope: :
The magic is that the tangent ratio of the rise over run 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
Determine The Roof Slope Multiplier
Note: If you already know the roof slope in degrees, then all you have to do to find the roof slope multiplier is find the secant using a scientific calculator.
For example, if the roof slope angle is 45°, then sec = 1.414213. That will be your roof slope multiplier.
So, to proceed.
Once you have the slope of the roof expressed in standard pitch, you can consider yourself in possession of the lengths of the a and b sides of a right triangle. Now you need to pull out your trusty Pythagorean Theorem and figure out the length of the hypotenuse.
A good calculator can be found at web2.0calc.com.
As Im sure you remember, a² + b² = c². So, if your slope is 6:12,
6² + 12² = c²
36 + 144 = c²
13.416 = c
So 13.416 is the length of the hypotenuse.
Now that you have the length of your hypotenuse, 13.416, you want to think of that as the length of a line on the actual sloped roof surface over a horizontal distance of 12 inches. You want to find out how much longer that line is than the 12 inches it covers.
You will express this difference as a number, called the roof slope factor .
Once you know the roof slope multiplier for any particular roof slope, you can use it to find the actual area of a roof of any size that has the same slope.
So in this case, for every 12 inches the roof goes across horizontally , while rising 6 inches , theres actually 13.416 inches of roof .
If you divide the 13.416 inches by the 12 inches , that gives you 1.118.
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