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Calculating Pitch and Rise With Algebra

By Kathleen Knowles, 30 Sep 2020

Roofing types, roofing designs, roofing materials, and roof categories are among the important considerations when constructing a building. When all these are settled, the roof's configuration, usually defined by the pitch and rise, needs to be calculated.

A roof is a structure that covers the top of a building or a vehicle. Roofs are required to prevent the ingress of unwanted elements. They provide protection from rain, snow, sun, high and low temperatures, and other unfavorable environmental conditions. They are also used as an aesthetic element for buildings. In modern-day houses and favorable climes, roofs are used as a means to generate electricity for the home (solar panels).

Some useful definitions

  • Rafter: This is a beam or collection of beams that make up the skeleton of a roof.
  • Run: Run is defined as the horizontal distance as measured from the roof's centerline to the edge of the roof.
  • Rise: Rise is explained as the vertical distance traveled per unit horizontal distance.

The terms defined can be used to represent the roof pitch. The pitch of a roof is the slope made by the framework of the roof. The slope can be represented as

  • The ratio of rise to run: This is one of the most popular ways pitch can be easily interpreted among architects, construction experts and building technicians. It is often represented as x/12 or x:12. For example, 5/12 pitch means that the rafter rises five (5) inches (or preferred measurement unit such as centimetres, yards etc.) for every twelve (12) inches of horizontal distance (run) from the centreline to the edge of the roof. In the same way, 1/12 pitch means that the rafter rises one (1) inch for every twelve (12) inches it runs. The denominator is usually 12.
  • Angle: The pitch angle is the elevation angle from the edge of the roof (lowest point) to the center of the roof (highest point). This represents the angle made between the diagonal rafter and the horizontal rafter or between the slope and the run. When the angle is provided, it can be used to obtain the rise of the roof.
  • Percentage: This is similar to the rise/run ratio and can be used interchangeably. It is, however, not as common as the rise/run ratio. To convert the rise/run ratio to a percentage, for example, 3/12, we multiply the ratio by 100%. That is 3/12 * 100% = 25%

Roof Categories Based on Pitch

  • High Pitched Roof: This type of roof can have a pitch of up to 20/12. High pitched roofs have steep slopes. They are usually employed to move water from rainfall or snow away from the roof in the fastest possible time. When pools of water form on the roof surface that is exposed, there is a greater probability that leaks can occur. High pitched roofs are also employed by individuals that desire additional space for attics.
  • Conventional Roof: This is one of the most popular roof types. They are easy to construct and relatively easy to maintain. Conventional roof pitches are usually between 4:12 and 9:12.
  • Low Pitched Roof: These types of roofs are difficult to maintain because of difficulty in drainage resulting in the build-up of water pools that can cause leakages and the build-up of debris. Their pitches are typically between 3:12 and 4:12
  • Flat Roofs: These roofs are not actually flat, as they have a slight slope that guides water or snow towards the drain. They have pitches between ½:12 and 2:12.

Generally, the integrity of the roofing can also be enhanced by the material used. Some roofing materials include metal, slate, clay and concrete tiles, asphalt, and solar tiles.

Calculating Pitch and Rise

To calculate the pitch and rise and other important parameters of a roof, three equations are required.

Using Pythagoras Theorem: (Rafter length)2 = Rise2 + Run2 … (Equation 1)

Using the definition of pitch: Pitch = Rise/Run … (Equation 2)

Using trigonometry when given the angle: Pitch = Tan(angle) … (Equation 3)


A fairly inexperienced roof technician was given the following parameters for the roof of a residential building: Rafter length = 130 inches, Run = 120 inches. Find the rise, the pitch, the angle, and possibly the category of roof used here.

Given the rafter length and the run, we can apply the Pythagoras Theorem to calculate the rise:

From equation 1 above, we have:

1302 = Rise2 + 1202

Rise2 = 16900 – 14400

Rise2 = 2500

Rise = 25001/2

Rise = 50 inches

To calculate the pitch, we apply Equation 2 above:

Pitch = Rise/Run

Pitch = 50/120

This can be simplified as:

Pitch = 5/12 or 5:12

From the pitch value above, we can conclude that the roof is of the conventional type.

To calculate the pitch angle, we apply Equation 3:

Pitch = Tan (angle (in degrees))

Angle (in degrees) = Tan-1(Pitch)

Angle (in degrees) = Tan-1(5/12)

Angle (in degrees) = 22.62o

A roof builder is required to erect a roof with the following parameters gotten from the architectural drawing: Rise = 9ft and Run = 12ft. Calculate the angle, the pitch, and the rafter length. Also, determine what category of roofs it will fall under.

From Equation 1, we can derive the rafter length:

(Rafter length)2 = 92 + 122

(Rafter length)2 = 81 + 144

(Rafter length)2 = 225

(Rafter length) = 2251/2

Rafter length = 15ft

From Equation 2, we can calculate the pitch:

Pitch = Rise/Run

Pitch = 9/12 or 9:12

We can infer from the pitch that the roof is the conventional type. Roof pitches above 9:12 are considered high pitched.

From Equation 3, we can calculate the angle

Pitch = Tan (angle (in degrees))

Angle (in degrees) = Tan-1(Pitch)

Angle (in degrees) = 36.87o

As demonstrated above, using the three equations, one can derive the important parameters from the already given ones. In modern times, specialized software is used to perform these calculations; however, it is important to know the bases from which they are derived.

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