Even if the floorplan of your house is a nice simple square, unless your roof is flat, it’ll be a little tricky to get right. Sure, there are several types of basic roofing styles that you can choose from, and some are certainly simpler to construct than others. But today we’ll be talking about hipped roofs.

Hipped roofs are perhaps the trickiest of the basic roofing types, due to the angles used for the hip rafters. Obviously, there is a range in complexity when it comes to hipped roofs, but we’ll stick to the simplest and most basic type: a square base and 4 identical sides that slope up to a single vertex – aka a “pyramid roof”.

Before we get started here, you should note that in many countries, hip rafters are not angled at its top face to directly accommodate plywood or roofing joists. What this means is that such hip rafters are sunk below the theoretical intersecting points of the joists or sheathing and are not used for fastening purposes.

Here in Japan however, the hip rafter – along with all jack rafters – is the component into which plywood sheathing is nailed (jack rafters are generally in the same plane as hip rafter), and is therefore angled at the top surface to accommodate both intersecting sides.

**Basic Geometry and Trigonometry Critical to Understanding Hip Rafters**

Having a basic understanding of right triangles is key to this phase of house-building. Why? Because roof slopes are best described and easily calculated using a right triangle, via methods such as the phythagorean theorem and trigonometric functions such as sine, cosine, tangent, and their inverse functions.

Here in Japan, we use what we call a sashigane, or Japanese carpenter’s square, to mark angles on lumber. More specifically, we use a customized sashigane with “*urame (裏目)*” inscribed on the backside. Urame marks the measure of the number multiplied by the √2.

**Laying the Groundwork – Isolating the Unknowns**

*Note that decimals are rounded to 2 or 3 digits. Add additional digits for increased accuracy.

Information we have (knowns):

Roof slope – 40 percent, or a rise over run of 4:10 (2:5).

Side length – 4 meters.

So we understand that the roof base is square, at 4 meters per side, and we have a roof side-slope of 40 percent. These are our “knowns”. Our “unknowns” are the following (click on link to jump to section):

- Jack rafter length and end-angles
- Jack rafter spacing on hip rafter
- Hip rafter length and slope
- Angles for hip rafter ends
- Angle for hip rafter ridge

**Jack rafter Length and End Angles**

The jack rafter length (from top point to bottom point) can be found using the **law of sines**, as we know the base length (2 meters, half of the side-length) and all 3 angles due to the **triangle postulate** – a 40 percent slope translates to 21.8 degrees as per the equation **slope = arctan0.4**. We also know that because the sine of 90 degrees (angle C) is 1, the length of the hypotenuse c is equal to 2 (side *b*) over the sine of 68.2 degrees (angle B).

*Note the distinction/relationship between the angle off 90 degrees and off 0 degrees. Table saws are measured off 90 degrees, where 90 degrees on the stock equals 0 degrees on the saw setting.

In this case the longest JRL = 2.154 meters minus the hip rafter end-point to edge distance (jump to image, note red line with star). Remember that this measurement is a theoretical tip to tip assuming a strict triangle, with no other geometry. In practice, you’ll most likely extend the rafters past the square base. Note that in our example here, the top end of each of the four center jack rafters must be angled on both sides from center, to fit between hip rafters (see image).

A common practice when jack rafters aren’t precut to size, is to go ahead and fasten them in their positions, letting them overhang, and cut them by hand after they’re up. There are ways to calculate each one, but being that the rafter ends need to line up more or less perfectly, and wood just isn’t always that accurate, cutting them after the fact is more common than you might think.

The alternative is to get them precut slightly shorter than the theoretical length and align the bottom ends of each rafter. What this means practically, is that any discrepancy will be brought to the top (better too short than too long in this case), as priority is placed on the rafter end alignment.

**Jack rafter to hip rafter connection angle**

Jack rafters meeting a ridge beam will simply need to be cut at 68.2 degrees, but those that meet a hip rafter, the angle will be diagonal – slicing through both horizontal and vertical axes in relation to the rafter sitting horizontally.

It’s clear from a bird’s eye view that the horizontal angle must be 45 degrees. On a miter saw this would be the “saw-head” angle. The second component of the angle – the “turn-table” angle – would be 22 degrees (21.8 to be more accurate). This is simply the degree-angle of a 40 percent slope.

Remember here that the turn-table angle **0 degrees** equals **90 degrees** on the stock. This means 22 degrees on the turn-table actually translates to 68 degrees on the stock – which probably makes more intuitive sense in your mind.

**Jack rafter spacing on hip rafter**

Our roof pitch is 40 percent, and we want 30 centimeter spacing between jack rafters center to center. Once the fascia is up, you can simply mark 30 centimeters for the bottom end. You can do this on the outer perimeter beams as well. The question here is the top end, or the spacing **on the hip rafter**. For this we incorporate a little geometric logic.

Imagine a horizontally oriented 45-45-90 triangle with opposite and adjacent sides of 30 centimeters each. Via the Pythagorean theorem we know the hypotenuse equals 42.42 centimeters. This however, does not equal the hip rafter spacing due to the hip rafter slope we haven’t yet taken into account.

We now take a vertically oriented right triangle situated such that its run (adjacent side) equals (sits on) the hypotenuse of our last triangle – 42.42 centimeters. It has a rise (opposite side) of 28.28 percent (the hip slope) of the run – 11.99 centimeters. Again, via the Pythagorean theorem we know that this new hypotenuse equals 44.08.

**44.1 centimeters is the hip rafter spacing.** By adding additional decimals to the calculation you can get more accurate values.

Another way to come to the same answer is:

Since the slope is 40 percent, we have to take the ratio of that hypotenuse (10.77cm) and the length of its run (10cm). We then scale this ratio up by a factor of 3 to obtain the correct hypotenuse for the distance on the hip rafter: (10.77×3)^{2}+(10×3)^{2}=44.1^{2}. Making your own custom ruler streamlines the process, and can result in higher accuracy.

*See table for all relevant right triangle side lengths. 40 percent slope is highlighted in light green.

The hip rafter slope is found by first multiplying half the base-length (2) by the √2, which gives us the adjacent side length, 2.828. We then use the pythagorean theorem to calculate the opposite side-length, which is the **same for both hip and jack rafters**, 2.154^{2}-2^{2}=0.8^{2}.

And we again use the pythagorean theorem to calculate the hypotenuse of the hip rafter, and from there use some trigonometry to compute the remaining angles. 2.828^{2}+0.8^{2}=2.939^{2}. As long as you have the rise (0.8) and run (2.828), you can convert the slope to a decimal and use the arctangent function to calculate the slope angle.

We use the good ol’ triangle postulate to get the remaining angle – 180-90-15.795=74.205 degrees. Having this angle is important when setting your table saw to cut the ends — although you’d have to make a jig, as I have yet to see a table saw that can cut more than about 45 degrees. You can however, bypass having to cut this sharp angle if your rafters will overhang past the exterior walls.

Since the roof for this example is pyramid-shaped, all 4 hip rafters will merge at the vertex. We’ll start with the bottom end. The bottom tip of the hip rafter should make a theoretical line with the bottom tips of the jack rafters in both sides (directions) adjacent to the given hip rafter.

We begin by marking the point on the center of the top surface of the hip rafter that will be the outermost point. Extend lines from this point perpendicular to the rafter edge. Place your sashigane onto the rafter so that it makes the same slope as the hip rafter (rise 0.8 : run 2.828), where the rafter edge is horizontal (see image below).

The dotted blue line equals half the width of the hip rafter, and is added to the length of the slope from the rafter-edge to the perpendicular line we drew earlier. This then determines the final position of the edge point. The red lines with the stars illustrates the top surface cut line.

We now have to mark out the side angle. Again, place your sashigane onto the side of the rafter so that it makes the same hip rafter slope, with the rafter top edge as the hypotenuse. The top edge of the sashigane should be aligned with the line we drew out in the previous step. Extend that line down the side of the rafter.

This line should be plumb in relation to the actual slope of the hip rafter. Now place your sashigane on the line we just drew so that its legs are aligned at the ratio 1.13:2, with the longer leg at the top (see image above). This ratio is the opposite side-length times the √2 and the adjacent side of the primary roof slope (not the hip).

Another way to do this is to draw a perpendicular line out from the plumb line, with the two legs forming the ratio 1:0.4x√2 – vertical and horizontal legs respectively. This ratio is the **roof rise times the √2 and run**. Now connect the point where the line meets the edge at the top and the point the horizontal line ends, and extend it through to the other side of the rafter.

Now do the same thing for the top end. This will result in all four hip rafters converging at the vertex. Bolting the rafters together as in this image, is a safe and advisable connection method. The image shows a 4 sided hipped roof, but with a suspended, central connection stock. You can probably use the same bolting method with or without the center stock, but having this notched central stock provides a solid base into which all the converging rafters can lock into.

There are 2 ways to achieve this angle. You can use one to double-check the other.

Method #1:

Begin by snapping an inkline down the center of the top surface of the rafter. Now draw out a “level line” on the side of the hip rafter by aligning the ratio 2.2828:0.8 to its top edge. Measuring from the edge, mark the distance half the width of the top surface of the rafter as shown in the image below.

By connecting the locus of all points this distance from the rafter edge to the inkline we snapped earlier, we can determine the angle at the rafter cross-section, and get a clear picture of the material that needs to be removed.

Method #2:

Take the right triangle that forms the hip rafter slope and scale it down to where the adjacent side equals the length of half the width of the hip rafter. Now take the altitude. This altitude equals the vertical distance from the top of the rafter cross-section, or opposite side of the right triangle that would fit into the section that is to be cut out for the ridge (see image below).

The final right triangle on the bottom left of the image above has been reconfigured slightly such that the adjacent leg (half-width of rafter) is on top, with the altitude taken earlier as the new opposite leg. It might seem – from looking at the image – that the ridge angle is equal to the hip rafter slope-angle.

However, it is always a slightly smaller angle, as you should notice intuitively if you examine the reconfiguration of the triangle from the earlier hip rafter slope, as well as the image below. 隅勾配 is the hip slope, 隅中勾 is its altitude, and 隅木山 is the resultant ridge slope.

You can trace this angle onto a piece of plywood, and use it to double-check your work, or take the angle using a protractor so you can set your saw.

Reference:

– Biglobe – What is a Sumigi?

– Monotsukuri – (規矩術技能; *kikujutsuginou*; compass & straightedge workshop)

– Monotsukuri – Japanese Woodworking/Angle Calculator.