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The roof pitch is a measure of roof steepness.  The pitch is defined as the slope of the roof, and is commonly represented as the ratio of the RISE over the RUN in the form x/12.

The rise is the height of the roof at its highest point, and the run is the horizontal span of the roof, measured from the roof ridge to the side of the building.

As an equation:

$$Pitch = {rise \over run}$$

For example, if a roof has a pitch of 2/12, then for every 12 inches the building extends horizontally, it rises 2 inches.  Similarly, if your roof has a total RUN of 24 feet and a RISE of 4 feet, then the pitch is:

$$Pitch = {4 \over 24} = {2 \over 12}$$

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## What is the roof pitch factor?

The roof pitch conversion factor is a number that, when multiplied by the area covered by the roof, gives an estimate the total surface area of the sloped roof itself.  It is sometimes called the roof pitch multiplier.

The formula for the roof pitch multiplier comes from the formula for a right triangle, and more specifically the Pythagorean Theorem:

$$Rafter^2 = rise^2 + run^2$$

Here the RISE and RUN are defined as above, and the RAFTER is the length of the hypotenuse of the triangle with side lengths RISE and RUN.

Now:

Solving for RAFTER in the above equation gives:

$$Rafter = \sqrt{rise^2 + run^2}$$

The roof pitch factor is then the length of the RAFTER divided by the RUN and is given by the following roof pitch factor formula:

$$Roof\,pitch\,factor = {rafter \over run}$$

Check out the conversion chart to find approximate roof pitch multipliers for different values of the pitch:

Roof Pitch Angle Roof Pitch Factor
1/12 4.76° 1.0035
2/12 9.46° 1.0138
3/12 14.04° 1.0308
4/12 18.43° 1.0541
5/12 22.62° 1.0833
6/12 26.57° 1.1180
7/12 30.26° 1.1577
8/12 33.69° 1.2019
9/12 36.37° 1.2500
10/12 39.81° 1.3017
11/12 42.51° 1.3566
12/12 45.00° 1.4142

## How is the roof pitch factor calculated?

Suppose you are interested in a roof with a pitch of 3/12.

We can use the Pythagorean Theorem to find the RAFTER length corresponding to a horizontal span of 12 inches and a rise of 3 inches:

$$Rafter = \sqrt{3^2+12^2} = \sqrt{153} = 12.37 \,in$$

Then the roof factor is found by dividing RAFTER by RUN:

$$Roof\,pitch\,factor = {12.37 \over 12} = 1.03$$

## How to find the surface area of a roof

So how do you use the roof pitch multiplier to find the surface area of a roof?

Let’s look at an example.  Assume that your roof covers a region that is 10 ft by 15 ft and the pitch is 3/12.   Scale the surface area by the multiplier found in the table:

$$Surface\,area = Area\,covered\,by\,roof × Roof\,pitch\,factor = 10 × 15 × 1.03 = 150 × 1.03 = 154.5\,square\,feet$$

## Step-by-step example using the roof pitch factor table

1. Measure the dimensions covered by the roof. For this example, suppose the roof is 20 ft wide (measured from the roof ridge to the edge of the building) by 40 ft long.  This means: $$Area\,covered\,by\,roof = 20 × 40 = 800\,square\,feet$$
2. Calculate the pitch. You can use this roof slope calculator.  Here let’s assume the pitch is 6/12.
3. Look up the multiplier in the table, or use the method outlined above. We calculate that a pitch of 6/12 corresponds to the factor 12.
4. The surface area of the roof is found by multiplying the number found in Step 3 by the number found in Step 1: $$Surface\,area\,of\,roof = Area\,covered\,by\,roof × Roof\,pitch\,factor = 800 × 1.12 = 896\,square\,feet$$

In other words, the surface area is impacted by the slope of the roof. A greater slope means a greater correction factor in the form of the roof slope multiplier.

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