Sunday, March 22, 2020

Using Pitch in Right Triangle Calculations

Using Pitch in Right Triangle Calculations

Using Pitch to Solve Right Triangle Lengths

In construction problems, we sometimes are working with roots and other structures that are shaped as right triangles.  In roofing applications, we are often working with the pitch of the roof.  The pitch is similar to the standard angle of right triangle.

The pitch is defined in 1 unit of rise over 12 units of run.  In the United States, the units are typically either feet or yards. 

pitch = 1 unit of rise / 12 units of run

With the right triangle, we can use similar triangles to determine that:

pitch /12 = rise / run

Knowing either one of the variables, we can use ratio calculations to determine the other.



Example 1:

Pitch: 3/12, Run:  48

Rise: 
12/48 = 3/x
x = 3 * 48 / 12
x = 12

Hypotenuse:
√(48^2 + 12^2) ≈ 49.4773

Example 2:

Pitch: 5/12, Rise: 30

Run:
5/12 = 30/x
x = 30 * 12 / 5
x = 72

Hypotenuse:
√(30^2 + 72^2) = 78

Approximating Angle with Pitch

To find the angle using pitch:

θ = atan(pitch / 12)



If you do not have a scientific calculator, you can approximate the angle by using any of the approximations (these are not the only approximation equations).   They were found using the curve fitting features of a Casio fx-9860gii.

θ ≈ 3.72674504 * p  + 2.78886014     (r^2 = 0.99084258)
θ ≈ 4.93070418 * p^0.91971679    (r^2 = 0.997149)
θ ≈ -0.1158911 * p^2 + 5.17538414 * p - 0.3498579   (r^2 = 0.9999772)

Data used:  θ to four decimal places are used. 

Eddie

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