Hip Drop, Mark & Back Bevel Angle

To calculate the drop of any hip you will need to use both the angles that the hip makes with the corner(cheek) and the pitch angle of the hip itself. In a single pitch roof or regular hip those angles would be 45° for the cheek, (this is the angle made by bisecting the 90° corner) and the hips pitch angle. This is the angle that the hip makes with the ground or level. When calculating a split pitch hip you will still need these angles, these hips will have two different intersecting angles and a common pitch angle. Using the two examples of 8/9 and 7/12 the angles would go like this.

8 side = 41.63° pitch angle 26.49°
9 side = 48.37° pitch angle 26.49°
7 side = 30.26° pitch angle 26.74°
12 side = 59.74° pitch angle 26.74°

The last determining factor in calculating the hip drop is the thickness of the hip member. You will always use ½ of its thickness to calculate the drop amount. Since in this case we’ll say that the Hip is 1-1/2" thick, that would make ½ of the thickness of the hip equal to 3/4" or .75".

If you were inclined to drop the Hip rafter as opposed to beveling its top edges this is the way to calculate the “extra” amount to remove from the seat cut of the birds mouth.

The equation is then;

(½ hip_thickness * tan(cheek)) * (tan(hip pitch angle))

For the 8 rise side (.75 * tan(41.63)) * (tan(26.49)) = .3322"
For the 9 rise side (.75 * tan(48.37)) * (tan(26.49)) = .4205"

For the 7 rise side (.75 * tan(30.26)) * (tan(26.74)) = .2205"
For the 12 rise side (.75 * tan(59.74)) * (tan(26.74)) = .6478"

The Bevel mark is the mark or amount that you would measure down from the top of the hip rafter to cut the hip back bevel. Since you cut from the center of the hip rafter to its outside edge, the bottom leg of the triangle that is formed is always equal to ½ the thickness of the hip. If you look at a section of the hip rafter you can see that if you draw a line from the bottom of the bevel mark to the top of the center line of the hip you will create the angle needed for the jack rafters to plane into the hip. You can calculate this amount by using the cosine function. Applying it to the amount of hip drop will give the correct amount to mark.

Using .2205" and .6478" respectively for the 7/12 pitch we can multiply the drop amount by the cosine 26.74°, which is the pitch angle of the hip.

This works out to;

.2205 * cos(26.74°) = .1969 or ~ 7/32" and
.6476 * cos(26.74°) = .5802 or ~ 19/32"

Using .3322" and .4205" respectively for the 8/9 pitch we can multiply the drop amount by the cosine 26.49°, which is the pitch angle of the hip.

This works out to;

.3322* cos(26.49°) = .2973 or ~ 5/16" and
.4205 * cos(26.49°) = .3764 or ~ 3/8"

You can easily determine the bevel angle in degrees for the hip rafter by using the ArcTangent Function. Knowing one leg of the right triangle is always ½ the thickness of the Hip and the other leg is the amount of the bevel mark you can simply apply the function to each Bevel Mark;

For the 7/12 pitch roof it’s. . .

.5802 / .75 = .7736. . . tan^-1(.7736) = 37.73°
.1969 / .75 = .2625. . . tan^-1(.2625) = 14.71°

For the 8/9 pitch roof it’s . . .

.3764 / .75 = .5018 . . . tan^-1(.5018) = 26.65°
.2973 / .75 = .3964 . . . tan^-1(.3964) = 21.62°

This graphic illustrates the relationship between the center line of the hip and the back bevel angle for the 7/12 pitch roof.