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Hip/Valley Angle Layout

Calculate exact common plumb, hip plumb, and cheek cut (bevel) angles for compound miter saw framing.

Roof Geometry

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Hip/Valley (16.97 Base)Common (12 Base)Ridge (4" Rise)

The 16.97 Hypotenuse 📐

Why is the Hip plumb cut totally different than the Common plumb cut?

A standard common rafter runs 12 inches across the ceiling. But a hip rafter branches out of a 90-degree corner exactly through the middle (45 degrees). Using the Pythagorean theorem $A^2 + B^2 = C^2$, we calculate $12^2 + 12^2 = 288$. The square root of 288 is approximately 16.97.

Therefore, the framing square layout for a hip is always [Rise] to [17], instead of [Rise] to [12].

Common Plumb Cut

18.4°
Standard Rafter Pitch

Hip Level Cut

76.7°
Seat / Birdsmouth

Hip/Valley Plumb Cut

13.3°

Miter Saw Angle 1 (Blade Tilt)

Hip/Valley Cheek Cut

71.6°

Miter Saw Angle 2 (Table Turn)

For estimation purposes only. Always consult a licensed professional before beginning work. Full Trade Safety Notice →
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What is Compound Roof Geometry?

Cutting a hip or valley rafter is one of the most complex tasks in framing because it requires making a 'compound cut'—a dual angle that leans forward (the plumb cut) while also twisting severely to the side (the cheek bevel cut) to mate against the corner of a ridge board.

Mathematical Foundation

Common Plumb Angle
θcommon=arctan⁡(Rise12)\theta_{common} = \arctan\left(\frac{\text{Rise}}{12}\right)
RiseRise= The vertical inches the roof rises for every 12 inches of horizontal run.
Hip Plumb Angle
θhip=arctan⁡(Rise16.97)\theta_{hip} = \arctan\left(\frac{\text{Rise}}{16.97}\right)
16.9716.97= The diagonal hypotenuse of a 12x12 square. Because a hip travels diagonally across the corner of a room, it travels 16.97 inches horizontally for every 12 inches the common rafter travels.
Hip Cheek (Bevel) Cut
Cheek=arcsin⁡(cos⁡(θcommon))Cheek = \arcsin(\cos(\theta_{common}))
CheekCheek= The side-to-side twist angle required to make the hip rafter mate flush into the corner of the ridge board.

Laws & Principles

  • The 16.97 Hypotenuse Rule: A standard common rafter steps across the ceiling exactly 12 inches at a time. A hip rafter, however, must span diagonally from the corner of a square room. Using the Pythagorean theorem (12^2 + 12^2 = 288), the square root of 288 is exactly 16.97 inches. Thus, the framing square layout base for all standard hips and valleys is 17.
  • The Trap of the False Bevel: A common mistake is assuming the side cut (cheek bevel) to mate the hip to the ridge board is a simple 45-degree angle on a miter saw. This is only true if the roof pitch is 0/12 (completely flat). As the roof pitch gets steeper, the horizontal mating angle actually sharpens rapidly.
  • Compound Usage: When cutting with a standard sliding compound miter saw, the saw table is physically spun left or right to match the Cheek Cut output angle, and the blade head itself is tilted downwards to match the Hip Plumb Cut output angle.

Step-by-Step Example Walkthrough

" Framing a steep 8/12 pitch roof and attempting to cut the hip rafter top-end to mate into a ridge block. "

  1. 1. Identify standard pitch: The common rafter plumb cut is atan(8/12) = 33.7 degrees.
  2. 2. Identify Hip pitch: Since the hip travels further (16.97 inches) to rise the same 8 inches, its slope is gentler. atan(8/16.97) = 25.2 degrees.
  3. 3. Determine the Side Bevel (Cheek): Take the cosine of the common plumb (33.7 degrees), and apply the arcsine function to find the cross-bevel.
  4. 4. Calculate: arcsin(cos(33.7)) = 56.3 degrees.
Final Result: To execute the perfect cut, the carpenter sets the miter saw blade tilt (bevel) to 25.2 degrees (Plumb) and spins the miter saw table over to precisely 56.3 degrees (Cheek / Side cut).

Quick Answer: How do you cut a hip or valley rafter angle?

To cut the top of a hip or valley rafter, you must make a compound miter cut. First, calculate the Hip Plumb Angle by dividing your roof Rise by 16.97 and taking the arctangent. You tilt the blade of your saw to this angle. Second, calculate the Cheek Cut (miter angle) using the formula arcsin(cos(Common Plumb Angle)). You spin the base of your miter saw to this cheek angle. Make the cut with both angles engaged simultaneously.

Hip & Valley Angle Formulas

Hip Plumb Cut Tilt = arctan(Rise / 16.97)

Cheek Cut Swing = arcsin(cos(Common Plumb Cut))

Hip Backing / Drop = arctan(sin(Common Plumb Cut))

Note: When carpenters refer to dropping a hip, it means making the plumb cut of the birdsmouth slightly deeper than the common rafter birdsmouth so that the edges of the hip (which stick up higher due to the bevel) sit flush with the plane of the common rafters.

Standard Pitch Cheek Cuts

Roof Pitch Common Plumb Hip Plumb (Blade Tilt) Cheek Angle (Table Spin)
4/1218.4°13.3°71.6°
6/1226.6°19.5°63.4°
8/1233.7°25.2°56.3°
10/1239.8°30.5°50.2°
12/1245.0°35.3°45.0°

Notice how a 12/12 pitch is the magical crossover point where the cheek angle finally exactly equals the common plumb angle. On all lower pitches, the cheek angle is surprisingly steep (often steeper than a standard miter saw can swing, requiring cutting on edge).

Construction Scenarios

The Flat Miter Illusion

A novice framer assumes that because the outside corner of the house is 90 degrees, the hip rafter must mate to the ridge exactly at 45 degrees. They spin their saw to 45 degrees and make the cut. The board does not fit. Because the board is leaning backward (the plumb curve), the geometry of the side-cut distorts. A 6/12 roof actually requires a massive 63.4-degree cheek cut to mate tightly against the ridge block, not 45 degrees.

The Miter Saw Limitation

A carpenter is framing a 4/12 roof. They calculate their cheek angle as 71.6 degrees. They go to their expensive 12-inch sliding miter saw, unlock the base, and realize the saw only spins to 50 or 60 degrees. Because the angle exceeds the mechanical limits of modern saws, the carpenter must stand the board vertically against the fence and use the blade tilt (bevel) to execute the 71.6-degree cut instead of the table spin.

Hip & Valley Framing Tips

Do This

  • ✓Back the Hip or Drop the Hip. A standard hip rafter will stick up slightly higher than the common rafters if left as a square block. You must either physically run a circular saw down both top edges to bevel them ("backing"), or make the birdsmouth notch slightly deeper to lower the entire board ("dropping").
  • ✓Use a layout square. If you do not have a scientific calculator, note that every framing square has a "Hip/Val" line stamped directly into the metal. The numbers on that line inherently handle the complicated tangent and cosine geometry for you.

Avoid This

  • ✗Never assume valleys and hips are different. From a pure geometric perspective, a valley is simply an inverted hip. The exact same 16.97 multiplier, plumb tilt, and cheek angle calculations apply identically to both members.

Frequently Asked Questions

Why do hip rafters use 17 instead of 12 for the run?

Because hips sit diagonally on the corner of a building. If a common rafter goes straight back 12 inches, and the adjacent wall goes straight over 12 inches, the diagonal line completing that square is the square root of (12² + 12²), which equals 16.97 inches. We round 16.97 up to 17 for simplification on steel framing squares.

What is the cheek cut on a rafter?

The cheek cut is the side-bevel required to make a rafter mate tightly against an opposing piece of material. On a hip rafter, the cheek cut allows the face of the board to twist and sit perfectly flush against the corner of the joining ridge boards.

Is a valley rafter cut exactly the same as a hip?

Mathematically, yes. Both traverse the 16.97-inch diagonal. The only physical difference is orientation. A hip represents an outside peak corner (water sheds away), while a valley represents an inside crease (water funnels into it). The angles themselves are identical.

What if my miter saw can't turn past 50 degrees?

Many lower-pitch roofs require cheek cuts over 60 degrees. Standard miter saws cannot swing this far. You have two choices: either cut the board by hand using a circular saw by tracing the line, or stand the board vertically against the saw fence and utilize the saw's blade-tilt function instead of the table-spin function.

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