Hip and Valley Factor Calculator, Table, and Guide

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Introduction

In this article, I’m going to explain what a hip and valley factor is, show you the math, and give you a pretty comprehensive table where you can look up the hip and valley factors for most roof slopes. I also created a calculator that lets you calculate the hip and valley factor for any roof slope, even slopes not included in the table. This article is not about how to cut rafters.

Important: The values discussed on this page and given in the hip and valley factor table only apply to regular hips and valleys, where the two roof sections have the same slope and meet at a 90 degree angle in the plan view (viewed from above). The information on this page will not apply to the hip rafters on an octagonal roof, for example.

A regular hip rafter and a regular valley rafter on the same roof typically have the same slope if both roof planes have the same pitch. This is why there is one hip and valley factor instead of a separate hip factor or valley factor.

Why They Share the Same Slope:

1. Diagonal Orientation: Both the hip rafter (at the external corner) and the valley rafter (at the internal corner) run diagonally across the roof plane.
2. Consistent Roof Pitch: If the roof planes share the same pitch, the rise-to-run ratio is the same for both the hip and valley rafters.
3. Geometric Symmetry: The diagonal relationship creates a consistent slope for both the hip and valley rafters since they are both oriented at a 45-degree angle relative to the main rafters.
4. Same Height and Distance Covered: Hip rafters and valley rafters on the same (regular) roof are at the same height since both rise the same distance from the eave to the ridge of the roof.

Hip and Valley Factor Calculator

What is a Hip and Valley Factor?

The hip and valley factor is a number that is multiplied by the run, or horizontal distance covered, of a common rafter to determine the length of a hip or valley rafter. Repeat, the horizontal distance covered, not the length, of a common rafter.

To use the hip and valley factor to effectively determine the length of a board that will be used as a rafter, you must account for the thickness of the ridge board, the birdsmouth, and any eave overhang.

Again, you can’t simply measure from the center of the roof to the outside of the wall plate and apply the hip and valley factor. Accommodation for the heel and seat cuts (the birdsmouth), the thickness of the ridge board, and any eave overhang should be taken into account when determining the actual length of the rafter.

The hip and valley factor varies according to the slope of the roof, as shown in the table below.

Hip and Valley Rafter Slope is Different from the Roof Pitch

On a related note, the pitch (properly the “slope”) of a hip or valley rafter will not be the same as the pitch of the adjacent roof sections. The slope of the hip or valley rafter will be lower than the slope of the adjacent roof sections.

This is because the hip or valley rafter has to rise the exact same total amount, from the height of the eaves to the height of the ridge, but it has to do it over a longer distance.

Where common rafters (the regular rafters) rise a certain distance over 12 inches, the hip or valley rafter will rise the same distance over 16.97 inches.

While the slopes of the common rafters are expressed as “X-in-12″, the slope of the hip and valley rafter on the same roof will be “X-in-16.97“.

So where two roof sections intersect to form a 90° angle (a regular hip or valley), and each roof section has, for example, a 6-in-12 slope, the hip or valley rafter at that intersection will have a slope of 6-in-16.97.

Expressing the same thing using degrees: the roof sections in the above example have a 26.57° slope, while the hip or valley rafter will have a 19.47° slope.

Remember that the heel cut, seat cut, and plumb cut (or ridge cut) for a hip and valley rafter will have angles that reflect this difference in slope. Do not cut them according to a template you have been using for the common rafters.

Hip and Valley Factor Formula (Standard Roof Pitch)

For a roof slope expressed as “X-in-12” (rise-in-run), the hip and valley factor is determined by finding the square root of ((rise/run)² + 2) for the slope of the adjacent roof sections.

Divide the rise by the run (the run is 12). Square the result. Add 2. Find the square root of the result.

Example:

Given a roof slope of 6-in-12, find the hip and valley factor:

Step 1: Express the slope as a ratio

The roof slope is given as 6-in-12, which simplifies to a ratio of:

\[ \text{slope ratio} = \frac{6}{12} = 0.5 \]

Step 2: Square the slope ratio

\[ 0.5^2 = 0.25 \]

Step 3: Add 2 to the squared result

\[ 0.25 + 2 = 2.25 \]

Step 4: Find the square root of the result

\[ \sqrt{2.25} = 1.5 \]

Result:

The hip and valley factor for a roof slope of 6/12 is 1.5.

Hip and Valley Factor Formula (Roof Slope in Degrees)

There’s a good scientific calculator online here if you want to try doing this.

For a roof slope expressed in degrees, the hip and valley factor is determined by finding the square root of the sum of the square of the secant of the slope angle and 1:

\[ \text{Hip and Valley Factor} = \sqrt{\sec^2(\theta) + 1} \]

where \( \theta \) is the slope angle in degrees, and \( \sec(\theta) \) is the secant of the angle, calculated as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This factor accounts for the additional diagonal distance covered by the hip or valley rafter compared to a common rafter.

Example:

Given a roof slope of 25 degrees, find the hip and valley factor:

Step 1: Given Angle in Degrees

\[ \theta = 25^\circ \]

Step 2: Calculate the Secant of the Angle

\[ \sec(25^\circ) = \frac{1}{\cos(25^\circ)} \] Since \( \cos(25^\circ) \approx 0.9063 \): \[ \sec(25^\circ) = \frac{1}{0.9063} \approx 1.103 \]

Step 3: Square the Secant Value

\[ \sec^2(25^\circ) = 1.103^2 = 1.217 \]

Step 4: Apply the Hip and Valley Factor Formula

\[ \text{Hip and Valley Factor} = \sqrt{\sec^2(25^\circ) + 1} \] \[ = \sqrt{1.217 + 1} = \sqrt{2.217} \]

Step 5: Take the Square Root

\[ \sqrt{2.217} \approx 1.489 \]

Result:

The hip and valley factor for a 25° roof pitch is approximately 1.489.

Roof Pitch Measurement Tools

If you want to verify the slope of your rafters to an amazing degree of accuracy and you like cool new tools, you should check out this digital level. It’s probably way too expensive for what you need, but it will tell you the rafter’s slope in degrees, rise/run, or percentage, and automatically convert from one to the other.

As a much cheaper alternative, I recommend this slope finder. It’s very inexpensive and very accurate.

One more thing: if you’re using this table, you should consider getting yourself a construction calculator. This one is very good.

Table: Hip and Valley Factors