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Roof slope definitions and illustrations (C) Carson Dunlop Associates Roof Rafter Length Calculation
How to calculate rafter length from roof rise or run & slope

Roof rafter length or calculation is easy if we know any two roof measurement numbers chosen from:

Roof total rise (at the ridge), roof horizontal run distance, rafter length, or roof slope in rise/run or slope or percent slope or even a tangent value.

These same calculations used for roof rafter length will let you compute the length of roof trusses or the length of ramp or stair support stringers.

This article series gives 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.

Page top sketch of various roof slopes is provided courtesy of Carson Dunlop Associates, Toronto.

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How to Calculate Roof Rafter Length from Slope & Rise

Calculate rafter length from roof slope and rise (C) InspectApedia.comQuestion: if my roof slope is 32° and the rise is 4' 9", how long will the rafter be

Anonymous asked:

The angle of elevation of a roof top is 32°. If the vertical change in height is 4 feet 9 inches as indicated, how long must the rafter be? - 2021/11/12

Moderator reply:

Here is a summary of how we use the TAN or tangent function to calculate the length of a rafter when all we know is the roof slope and the roof total rise.

Tangent of an angle is defined as the ratio of the length of the right triangle's side adjacent to the angle (a) to the length of the side opposite the angle (b).

Knowing the tangent of the roof angle or slope + the length of either of the two sides (roof rise or roof run) lets us calculate the length of the missing side.

Knowing the length of any two sides of a right triangle lets us calculate the length of the missing side, because the relationship of the three sides of a right triangle can be written as the square of their lengths/

[Click to enlarge any image]

Note: if your roof slope is expressed in the more widely used "rise" such as a rise of 7.5" meaning the roof slope rises 7.5" for every 12" (or foot) of horizontal run, and if you want to convert that to a roof angle in degrees (as we use roof angle on this page) then

see CONVERT ROOF SLOPE or ANGLE in DEGREES to RISE or RUN

Details of Rafter Calculations from Roof Slope

Here are the details of the rafter length calculation given above

THANK YOU for asking a helpful question: how to calculate rafter length if all we know is the roof rise and roof angle or slope.

In our sketch (above) and below in more detail we show you how to calculate the rafter length you want.

At and also at ROOF SLOPE CALCULATIONS in a section titled "How to Convert Slope or Angle in Degrees into Distance Rise or Run" we go through the basic calculations using a 38° roof.

You'd use 32° instead of our example of 38° roof slope and use the TAN function (or a TANGENTS TABLE like the one we provide in that article) to calculate the sloped side or upper side of your triangle - the rafter.

When we know the length of just one side (rise or b in your problem an din our sketch above)

and where we know the roof slope in degrees (32° )

we can use the Tangent function to give us the ratio of two sides of our roof triangle (one of which, the rafter length) is unknown.

Then we can use simple algebra and arithemetic to calculate the rafter length.

Here's how we do it.

In ROOF SLOPE CALCULATIONS we state

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.

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 your problem statement the tangent T of the 32 degree angle (the roof slope) is the ratio:

TAN (Roof Angle or Slope) = Roof Rise / Roof Horizontal Run

So we write the tangent of angle A as:

Tan <a = rise (b) / run (a)

or

tan 32° = rise (4 ft 9 in or all in inches: 48+9 = 57 inches) / run (unknown)

or, looking up 32 in the TAN table found at ROOF SLOPE CALCULATIONS (or on your calculator)

Tan 32° = 0.6249 = 57 / run (b)

using basic rules of algebra we can rewrite the equation: multiply both sides by run (b)

0.6249 x (b) = 57

b = 57 / 0.6249

b = 91.2 inches - this is the run length

Now that we used tan to find the run length, we’ll go back to our formula for the lengths of sides of a right triangle -your roof.

Key concept: If we know the length of any two sides of a right triangle we can calcluate the length of the missing side.

Remember the basic formula for the lengths of sides of a right triangle (half of a roof)

For any right triangle: a2 + b2 = c2

where for a roof we can set

Run2 + Rise2 = Rafter-length2

91.22 + 572 = rafter length2

whip out my handy smartphone calculator and punch the numbers

8317.44 + 3249 = rafter-length2

11,566 = rafter-length2 or the rafter length squared

whip that calculator again to give us the square root:

sqrt of 11,566 = 107.5 inches ! that’s your rafter length

In feet that's a run of 8.96 ft or nominally 9 ft.

Watch out: you will need to increase the rafter length (increase the "run" of the roof) by the amount of overhang that you want - how far the rooftop should extend out over or past the building exterior wall.

For help using measurements of slope, angle, or degrees to calculate roof rise or run lengths

see CONVERT ROOF SLOPE or ANGLE in DEGREES to RISE or RUN

Below: another useful illustration showing the relationship between roof rise & run & slope.

Watch out: the first Y1 value in the sketch is 10" not 12" but in normal roof framing and stair framing talk, the "unit rise" means the amount of gain in height over 12-inches or one foot of horizontal distance traveled.

We used 10" here to show that we al do these slope and rise or run conversions for any number, and that this procedure works for stairs as well as roofs.

How to convert a stair aingle to rise & run using tangent (C) Daniel Friedman

Good luck on your test, and thank you once again to my high-school algebra teacher.


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