1. One Mathematical Solution

2. Virginia Beach, Virginia October 2018 Timber Framers Guild Conference

3. This roof was the subject of a two day workshop in Madison last year
The roof frame was sold at the TFG Auction

4. The Given Information
The building has 4 square corners and all framing is symmetrical and plumb.
The long eave has a roof pitch of 14/12, the short eave has a roof pitch
of 20/12
Dimensions are given in feet and decimal inches
The dimensions shown are typical plan view information
All rafters will have the same amount of material measured plumb at the
outer edge of plate.
In many parts of the US this called the Height Above Plate or HAP.
There are other names for this dimension, do you have one?

5. As a timber frame carpenter building a hip roof we need to know
THE GIVEN (in our example):
Plan view dimensions (plates and ridge are level in this exercise)
Plan view dimensions of overhang (eave line will be level in this exercise)
Angle between eaves (90 degrees in our example)
Roof Pitches
Timber sizes (S4S FULL DIMENSION)
THE UNKNOWN
Rafter lengths
Plumb cut
Level cut
Sheathing cuts
Hip rafter plumb cut, level cut, length, cutting angles , jack rafter or purlin locations, plate locations, ridge location.
Hip Rafter Fascia cuts
THE PROCESS
Basic and intermediate drafting defines the problem
Basic math using a carpentry vocabulary and your favorite calculator gets us closer.
Orderly record keeping makes sense of the many steps and operations along the way

6. Note the dimension location of the jack rafters
Note the dimension location of the jack rafters. Dimensioning to the short point face is a learned personal preference with a practical sensibility.
In nailed-on work the subtlety is lost but when joinery is involved I always look to fill the housing/ mortice by working the clip. The layout point (working point) on the hip rafter is an extension of the short point length. Opposing jacks may not align on the hip length in irregular work.

7. We begin without numbers
Rafter a
rise
rise
Run a
This is a 2d graphic representation of some of the initial planes of our model.
Rafter m
Run m

8. Then we add the numbers 20 14 12
Put some numbers to the problem 12

9. Now we calculate the hypotenuse length 20 14 12
Do the math. 12

10. The geometric element that defines roof math is the right triangle.
One property of the right triangle we rely on is SIMILAR TRIANGLES.
In any triangle drawing a line parallel to any side results in a triangle that is similar to the original.

11. Importantly in our work a right triangle divided by a line at right angles(90 degrees) to any side also produces a similar triangle.

12. The mathematical concept related to similar triangles that simplifies roof math
is known as PROPORTION
PROPORTION is a statement that two ratios are equal
In our example
the ratio 14/12 = common rise/run m = /1
The ratio 20/12 = common rise/ run a =
The ratio rafter m/ run m= /12 =
The ratio rafter a / run a = / 12 =
In our example we could determine the slope angle that we have expressed as rise/run.
Do we need to know the numerical value of the angle?
Many of us use a layout tool that is dimension based not angle based.
If you use a protractor to layout and that tool is accurate to ¼ degree
then the angular dimension may have value
Roof math is concerned with similar triangles of unknown sizes
Knowing the proportion and some crucial plan view information we can determine unknown lengths.
Keeping track of the various relationships is critical to success.

13. From our earlier math 20 14 12
Do the math. 12

14. Now displayed as ratios
/ 12 =
20 / 12 = 1.667
14 / 12 =
12/12 =1
This means for every unit run we can calculate these other lengths. That is great but our roof pitches must have a common rise.
12/12 = 1
/ 12 =

15. Our hip roof is defined as two pitches: 14/12 main roof and 20/12 adjacent roof
By Definition: Both roof pitches rise to a common height
The following image describes the process of resolving the two pitches to a common rise.
This image then allows us to consider the run of each pitch to reach an equal height.

17. Using similar triangles we can do the math for the common rise.
By convention I use the 14 (lesser of the two rises) rise of the 14/12 roof as the common rise.
By similar triangles and the proportion 20/12 = 14/X
X=(12*14)/20
X = 8.40

18. 20 14 14
8.40
By similar triangles 20/12= 14/8.4 12

19. The diagonal line in this new drawing is the run of the hip14 14
8.4 12
We simplify the drawing

20. We can calculate the length of the hip run14 14
8.4 12
We simplify the drawing

21. Now we do the math to get to the unit geometric ratios
14/12 = / 1
14/ 8.40 = 1.667/.7
1.1667
1.1667 .7
1.2207 1

22. We also need to know the rafter lengths
Main Run = 1, Common Rise = , Main Rafter Length =
Adjacent Run = .7 , Common Rise = , Adjacent Rafter Length =

23. Add the rafter lengths to the image
1.3606
1.1667
1.1667 .7
1.2207 1
1.5366

24. The Common rise expressed graphically as part of the hip rafter triangle. The calculated hip length is shown
1.1667 .7
1.6885
1.2207
The common rise and the hip run yields hip length. This is the hip elevation plane 1

25. Using a specialty calculator we can solve for rafter lengths if both the pitch and run are known
Using any calculator we can solve rafter lengths for any run using the ratio of rafter length to run.
In our irregular pitch regular plan hip roof example there are many ratios that we can use to calculate, layout and fabricate the various pieces of our roof.
We have the basic various ratios and can further extrapolate useful relationships with these ratios.

26. From the plan view drawing we can calculate 14 pitch common rafters using
Plan view dimensions and the ratios we have defined

27. For 14 pitch common rafters in our exercise we can calculate the length from our ratio and the plan view run dimension
1.3606
1.1667
1.1667 .7
1.2207 1
1.5366
From the overhang to center line of building = 12’/2=6’ run
6’ * = ’ or 72” * = ”
From the overhang to the plate line 1’* = ’
Or 12” * = ”

28. Note the location of the 14 pitch rafter is measured
parallel to the run of the 20 pitch rafter
Do we have enough information to calculate the cut length of the jack rafters?

29. We do have enough information to calculate the jack length to the hip length line
This does not get us what we need to cut jack rafters because we have not allowed
for the hip rafter width.
Let’s look at the plan for more information
The hip rafter does not appear to be centered on the hip corner
Because the roof is not equal pitches and the HAP is the same for all rafters we must shift the width of the hip so the HAP is equal on both sides of the hip.
We have previously determined the direction of hip rafter
We can use this information to determine the hip shift.

30. The graphic solution is straight forward
Draw a line equal to the width of the hip rafter and square to the run
of the hip rafter on each side of the run and through our hip corner.
The dotted line indicates equal lengths.

31. Draw a line parallel to the opposite eave from the outer end of each line
Draw a line to connect the points where these lines cross each eave
Note the similar triangles and that this new line is the same length
as the width of the hip.

32. Using the width of the hip in plan view we draw the sides of the hip
Now we can calculate the amount of hip rafter that needs to be subtracted from the jack rafter run IN PLAN VIEW
NOTE the graphic demonstration. The point where the hip width crosses the hip run has the same elevation in each roof.

33. We revert to the base rectangle and the known hip width = 4”

34. By similar triangles using the ratios:
4”/ x = / 1
X=4 / =

35. 4 / y = / .7
(4 * .7) / = y
Y = 2.938

36. Now we need to solve for the two dimensions that add up to 4”
/ = XX / 1
XX =

37. Next / = YY / .7
( / ) * .7 = YY
YY =

38. Solving for the length along the 14 pitch run
2.6844/ .7 = XXX /
( / .7) *
XXX =

39. YYY / = / 1
YYY = *
YYY =

40. Rafter run to face of hip on 14 side = run to hip line – 4.6812
8.4 12

41. We now convert our work into the roof plane dimensions
I like to layout rafters and jack rafters to the hip or valley working point, this simplifies the
Double checking process, In My Opinion.
From the plan view reductions of the jack rafter length from working point to face
we then calculate
14 pitch run reduction converted to rafter length reduction
4.6812” * = ”
20 pitch run reduction converted to rafter length reduction
1.6057” * = ”

42. The ratio of the main length to adjacent run will allow us to solve for the 14 pitch jack rafters.
1.3606
1.1667
1.1667 .7
1.2207 1
1.5366
1.5366/.7= units of length per unit of run 20 pitch

43. The Math for #1 14 pitch Jack rafters
Run of the 20 roof = 29.20”
29.20 * = ” to WP
64.097” – ” = ”

44. The ratio of the adjacent length to main run will allow us to solve for the 20 pitch jack rafters.
1.3606
1.1667
1.1667 .7
1.2207 1
1.5366
1.3606/1= units of length per unit of run 14 pitch

45. The Math for #1 20 pitch Jack rafters
Run of the 14 roof = 22.25”
22.25 * = ” to WP
” – ” = ”
Repeat as needed

46. Now consider the layout points along the hip

47. A review of the basic information 14 14
8.4 12
We simplify the drawing

48. The Common rise expressed graphically as part of the hip rafter triangle. Use the original dimensions. 14 14
8.4
The common rise and the hip run yields hip length. This is the hip elevation plane 12

49. Ratio hip length to 14 pitch run = 1.6885/1 = 1.6885
The Common rise expressed graphically as part of the hip rafter triangle. Now the ratios
1.1667 .7
1.6885
1.2207
The common rise and the hip run yields hip length. This is the hip elevation plane 1
Ratio hip length to 14 pitch run = /1 =
Ratio hip length to 20 pitch run = / .7=

50. The hip run measured on the 14 pitch run is 1’ to the plate corner and
6’ to the peak

51. The ratio of 14 pitch run to hip length is 1.6885/1
We have jack rafter working points
The 14 pitch jack runs as measured on the 20 pitch run.
The working point along the hip is then 20 run * 20 HL ratio
29.20” * = ”
The 20 pitch jack rafter #1 runs 22.25” as measured on the 14 pitch run
22.25 * = ”
#2 20 pitch JR runs on the 14 pitch run
41.75 * =
#3 20 pitch JR runs on the 14 pitch run
61.25 * =

52. The same numbers define the layout angles for the various parts:
14/12 Plumb and Level cut commons and jacks
20/12 Plumb and Level cut jacks
14/ plumb and level cuts on hip

53. Jack rafter cuts as laid out on the roof surface are the same angle as
the sheathing cut and purlin top cut. Knowing which way to apply the angle is key. For jacks the cut use Rafter length: Eave length 14 14 12
We simplify the drawing
8.4
The jack cut on the 14 pitch side of the hip is / 8.4
The jack cut for the 20 pitch side is / 12

54. Graphic development of 14 pitch jack cut and sheathing plane
Hip Length
Rafter Length
Eave length

55. Graphic development of the 20 pitch jack cut and sheathing plane
Rafter Length
Eave Length
Hip Length

56. Following the same principles we can establish ratio based relationships between the sheathing planes, ground planes, elevations planes.
Note the purlin top cut layout is 90 degrees to the jack rafter layout

57. 12
8.4
To consider the hip rafter plumb cuts as laid out on the unbacked surface
We first extend the common runs until they intersect a line drawn square to the Hip run and do some math.

58. We need to solve new lengths to develop some angles
Similar triangles help us
Both new triangles are related to the original plan view
On the 14 pitch side of the roof
8.4 / 12 = 14 tangent / hip run
Solve for 14 Tangent
14 tangent = (8.4 * ) / 12
On the 20 pitch side of the roof
12 / 8.4 = 20 tangent / hip run
20 tangent = (12 * ) / 8.4
Now we have lengths we need to consider a length at right angles to these lengths that represents the Hip Plumb cutting angles

59. Look at the plan view
To generate the plumb cutting angles using the tangent lengths we need to introduce the hip length .
This is shown graphically as follows

60. Using the lengths we calculated we now have the angles for the hip plumb cuts

61. We have two more sets of angles to consider
The hip rafter tail, cut in the plane of the square cut common rafters is not the same layout as a plumb cut tail.
We need to develop the cuts on the top unbacked surface of the hip and the cuts on each side.

62. We begin with the basic information

63. We draw a line square to the 20 pitch rafter length that intersects with an extension of the 20 pitch run

64. Then we draw a line square to the 14 pitch rafter length that intersects with an extension of the 14 run

65. Now we draw :
A line that is the extension of the hip run
A line that is square to the hip run through the hip corner
A line square to the extension of each common run through the points created by the lines previously drawn square to the common rafter lengths.
These last lines represent the bottom of the purlin plane

66. Next we need to bring the true length of the 14 counter pitch square to the line
that represents the purlin length this allows development of
the true shape of the hip rafter cutting planes on the purlin face

68. Repeat the exercise

69. Complete the development for the 20 pitch roof

70. Draw the hip elevation triangle

71. Extend the rafter run lines to the Tangent line

72. Rotate the hip length to the ground, square to the tangent line

73. Connect the lines as shown.
These four lines are true length and angle on the plane of the unbacked hip.
Note the significant variation between the plumb and square to pitch development, a result of the steep pitches of our example.

74. You can do the math to establish the ratios.
The various angles are labeled to allow you to compare with the
Hawkindale sheet.
Note the layouts on the hip elevation.
Subtract 90 to find the angle in the Hawkindale sheet