Construction Calculator
Hip Roof Calculator
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Work out every dimension of a hip roof from the plan and pitch: ridge length, common and hip rafters, jack rafters with their common difference, total surface area and the cut angles — in feet and inches or metres.
Hip Roof Calculator
Rafters, ridge, area and cut angles
Results
Common rafter
17.89 ft
17′ 10 11/16″
Hip rafter
24.00 ft
24′ 0″
Cut Angles
The plumb cut is given both ways because both references are used on site; the two readings are complementary.
Jack Rafters — Longest First
| # | Run (ft) | Length (ft) | ft / in |
|---|---|---|---|
| 1 | 13.667 | 15.280 | 15′ 3 3/8″ |
| 2 | 12.333 | 13.789 | 13′ 9 1/2″ |
| 3 | 11.000 | 12.298 | 12′ 3 9/16″ |
| 4 | 9.667 | 10.808 | 10′ 9 11/16″ |
| 5 | 8.333 | 9.317 | 9′ 3 13/16″ |
Each jack is 1.4907 ft shorter than the last — the common difference, equal to spacing × pitch factor.
These are geometric figures only. Rafter sizes, spans, loads, bearing and connections are separate structural questions, and cut lists should be confirmed against a physical layout before cutting stock.
The run is half the shorter side (30 ft), not half the width, because all four planes of an equal-pitch hip roof rise to the same ridge. The ridge then lies along the longer side and measures the difference between the two dimensions.
The hip overhang is longer than the common overhang — 1.5000 against 1.1180 ft — because a hip rafter extends diagonally across the corner. The ratio is √(2 + m²) ÷ √(1 + m²) where m is the rise over run, which is 1.3416 here and always above 1. Ordering hip stock to the common overhang length leaves it short.
The hip sits at 19.471° against the common rafter's 26.565°, because a hip covers 16.97 units of horizontal run for every 12 the common covers — the carpenter's "17 inch" rule, 16.97 in to be exact.
Cut angles depend on the reference you measure from, so both are given. The plumb cut is 26.565° from vertical, which is the same as 63.435° from a line square across the rafter. The seat cut is 26.565° from the rafter edge, and the two cuts are complementary.
Jack rafters shorten by exactly 1.4907 ft each step — the common difference, equal to the spacing × pitch factor. Each of the four hips carries jacks on both sides, so there are eight jack runs in total.
Where a jack meets a hip it needs a 45° side cut in plan, since the hip bisects a square corner on an equal-pitch roof. That bevel is in addition to the plumb cut.
Roof area is the footprint × pitch factor, which is exact rather than an approximation: the four planes cover the plan once. Decomposing the roof into two trapezoids and two triangles gives the identical figure. Formulas of the form 2(length × rafter) + 2(width × rafter) double count and can overstate the area by most of its own value.
The overhang adds 161.00 sq ft of roof surface, 10.7% of the total. Area figures here exclude ridge and hip caps, valley flashing and waste, so add a covering allowance on top.
Step-by-Step Calculation
Step 1 — Orientation and run
Plan is 40 × 30 ft
The ridge runs along the longer side (40 ft), and the run is half the shorter side.
Run = 30 ÷ 2 = 15 ft
Step 2 — Pitch
Pitch 6/12 → angle = arctan(6 ÷ 12) = 26.5651°
Pitch factor = √(1 + (6 ÷ 12)²) = 1.118034
Rise = 15 × 6 ÷ 12 = 7.5 ft
Step 3 — Ridge length
Ridge = 40 − 30 = 10 ft
Step 4 — Common rafter
Without overhang: 15 × 1.118034 = 16.7705 ft
With overhang: (15 + 1) × 1.118034 = 17.8885 ft
The eave projects 1 ft horizontally, which is 1.1180 ft measured along the slope.
Step 5 — Hip rafter
A hip crosses the corner diagonally, so its horizontal run is the run × √2.
Hip run = 15 × √2 = 21.2132 ft
Hip rafter = √(21.2132² + 7.5²) = 22.5000 ft
Equivalently 21.2132 × 1.060660, the hip run times the hip pitch factor.
Hip pitch factor = √(1 + (7.5 ÷ 21.2132)²) = 1.060660
With overhang: (15 + 1) × √2 × 1.060660 = 24.0000 ft
The overhang continues down the hip plane, so it gains rise as well as run.
Hip slope angle = arctan(7.5 ÷ 21.2132) = 19.4712°, shallower than the common.
Step 6 — Roof area
A hip roof covers its plan once, so the sloping area is the footprint × pitch factor.
Footprint incl. overhang = 42 × 32 = 1,344.000 sq ft
Roof area = 1,344.000 × 1.118034 = 1,502.64 sq ft
Result
Ridge 10.000 · Common 17.889 · Hip 24.000 ft · Area 1,502.6 sq ft
Hip Roof in Plan View
On an equal-pitch hip roof all four planes rise to a single ridge. The shorter side sets the run (half its length), the hips run at 45° in plan from each corner, and the ridge is what is left of the longer side.
How Hip Roof Geometry Works
A hip roof slopes on all four sides, so instead of gable ends it has two more triangular planes meeting the main pair along diagonal hips. Everything about it follows from two numbers: the plan dimensions and the pitch.
The key insight is that the run is half the shorter side, not half the width. All four planes rise at the same pitch to a common ridge, so it is the shorter dimension that sets how far each rafter travels horizontally — and the ridge then lies along the longer side, measuring exactly the difference between the two. Swap the length and width in your head and nothing about the roof changes.
From there the pitch factor does the work. It is the ratio of sloped length to horizontal run, √(1 + (rise ÷ run)²), which for a 6/12 pitch is 1.118034. Multiply the run by it for a common rafter; multiply the footprint by it for the roof area. Both are exact.
Hip Roof Formulas
1. Pitch Factor and Run
2. Ridge and Common Rafter
3. Hip Rafter
4. Roof Area
5. Jack Rafters
Why Roof Area Is Just Footprint × Pitch Factor
A hip roof covers its own plan exactly once. Every square foot of footprint sits under precisely one square foot of roof plane, tilted by the pitch — so the sloping area is the footprint scaled by the pitch factor, and that is exact rather than an estimate.
You can check it the long way. Split the roof into its two trapezoids and two triangles, work each out on the slope, and add them up: for a 40 × 30 ft plan at 6/12 the decomposition gives 1,341.64 sq ft and footprint × pitch factor gives 1,341.64 sq ft. They agree to the last digit, and they agree for every plan and pitch.
This matters because a commonly circulated form, 2(length × rafter) + 2(width × rafter), double counts badly. On the same roof it returns 2,504 sq ft — 1.67 times the true figure. It is a formula shaped for a box, not for four planes that share a ridge.
The Hip Rafter Is Longer and Shallower
A hip crosses the corner diagonally, so it covers 16.97 units of horizontal run for every 12 a common rafter covers — the carpenter's "17 inch" rule. That makes it longer but less steep:
| Pitch | Common angle | Pitch factor | Hip angle | Hip factor |
|---|---|---|---|---|
| 3/12 | 14.036° | 1.030776 | 10.025° | 1.015505 |
| 4/12 | 18.435° | 1.054093 | 13.263° | 1.027402 |
| 5/12 | 22.620° | 1.083333 | 16.416° | 1.042500 |
| 6/12 | 26.565° | 1.118034 | 19.471° | 1.060660 |
| 8/12 | 33.690° | 1.201850 | 25.239° | 1.105542 |
| 10/12 | 39.806° | 1.301708 | 30.509° | 1.160699 |
| 12/12 | 45.000° | 1.414214 | 35.264° | 1.224745 |
The same diagonal geometry means the hip overhang is longer than the common overhang. A 1 ft eave projection extends a common rafter by 1.118 ft along its slope at 6/12 but extends the hip by 1.500 ft. The ratio is √(2 + m²) ÷ √(1 + m²) where m is the rise over run — 1.3416 at 6/12, 1.2247 at 12/12, and always greater than 1. Cutting hip tails to the common overhang length leaves them short.
Three Things Worth Getting Right
- Which side the ridge runs along. The run is half the shorter side. Taking half the width when the width is the longer dimension oversizes every rafter.
- The hip overhang. It is longer than the common overhang by a factor that depends on pitch, because the hip tail projects diagonally.
- Which reference your cut angle uses. 26.565° from vertical and 63.435° from the rafter edge describe the same plumb cut at 6/12. Check what your square is set to.
Jack Rafters and the Common Difference
Jacks run from the top plate up to a hip, getting shorter as they approach the corner. The amount they shorten by is constant — the common difference, equal to the rafter spacing × pitch factor. At 16 in centres on a 6/12 that is 1.4907 ft per step, so once you know the longest jack the rest follow by subtraction.
Each of the four hips carries jacks on both sides, which gives eight jack runs on a hip roof. On a 40 × 30 ft plan at 16 in centres that works out to 11 jacks per hip side, 88 in total, plus the commons along the 10 ft ridge.
| Jack | Run | Length | ft / in |
|---|---|---|---|
| 1 | 13.667 | 15.280 | 15′ 3 3/8″ |
| 2 | 12.333 | 13.789 | 13′ 9 1/2″ |
| 3 | 11.000 | 12.298 | 12′ 3 9/16″ |
| 4 | 9.667 | 10.808 | 10′ 9 11/16″ |
| 5 | 8.333 | 9.317 | 9′ 3 13/16″ |
Jacks need a 45° side cut in plan where they land on the hip, since on an equal-pitch roof the hip bisects a square corner. That bevel is in addition to the plumb cut.
Example Calculations
A standard hip, a steeper one measured to the plate, and a square plan:
Example 1 — 40 × 30 ft at 6/12 with a 1 ft Overhang
The standard case: ridge along the length, eaves projecting 1 ft all round.
Run = 30 ÷ 2 = 15 ft (half the shorter side)
Rise = 15 × 6 ÷ 12 = 7.5 ft
Pitch factor = √(1 + (6 ÷ 12)²) = 1.118034
Ridge = 40 − 30 = 10 ft
Common rafter = (15 + 1) × 1.118034 = 17.889 ft → 17′ 10 11/16″
Hip run = 15 × √2 = 21.213 ft, hip rafter = 22.500 ft to the plate
Hip with overhang = 24.000 ft exactly → 24′ 0″
Roof area = 42 × 32 × 1.118034 = 1,502.6 sq ft
Example 2 — 36 × 24 ft at 8/12, No Overhang
A steeper pitch measured to the top plate only.
Run = 24 ÷ 2 = 12 ft, rise = 8 ft
Pitch factor = 1.201850, slope angle 33.690°
Ridge = 36 − 24 = 12 ft
Common rafter = 12 × 1.201850 = 14.422 ft → 14′ 5 1/16″
Hip run = 16.971 ft, hip rafter = 18.762 ft → 18′ 9 1/8″
Hip slope angle = 25.239°, shallower than the common 33.690°
Roof area = 36 × 24 × 1.201850 = 1,038.4 sq ft
Example 3 — Square Plan: a Pyramid Hip
30 × 30 ft at 6/12. Equal sides, so the ridge vanishes.
Ridge = 30 − 30 = 0 ft — the four planes meet at a point
Common rafter = 15 × 1.118034 = 16.771 ft → 16′ 9 1/4″
Hip rafter = 22.500 ft → 22′ 6″ (four of them, one per corner)
There are no common rafters along a ridge, only hips and jacks
Roof area = 30 × 30 × 1.118034 = 1,006.2 sq ft
Cross-check as four triangles: 4 × ½ × 30 × 16.771 = 1,006.2 sq ft
At 16 in centres: 11 jacks per hip side, 88 in total
Before You Cut Stock
These are theoretical centre-line lengths. Real rafters need shortening for half the ridge thickness at the top, and hip rafters need shortening for the ridge and for the way they meet the corner — typically half the diagonal thickness of the ridge, which is more than the common rafter allowance. Add the birdsmouth seat and check the heel height works with your wall and fascia detail. The figures also assume an equal pitch on all four planes and a rectangular plan with square corners; an unequal-pitch hip, a bastard hip, or a plan that is slightly out of square all change the hip angle and the 45° jack side cut. Cut one rafter, offer it up, and confirm it before cutting the rest — a pattern rafter checked in place is worth more than any calculation. Note too that this covers geometry only: rafter sizes, spans, loads, bearing and connections are separate structural questions, and roof area excludes ridge and hip caps, flashing and waste, so add a covering allowance on top. This is general guidance, not a substitute for professional advice.
Frequently Asked Questions
- How do you calculate hip roof rafter length?
- Multiply the run by the pitch factor. The run is half the shorter building dimension, and the pitch factor is √(1 + (rise ÷ run)²). For a 30 ft wide building at 6/12, the run is 15 ft and the pitch factor is 1.118034, giving a common rafter of 16.771 ft to the top plate. Add the overhang to the run before multiplying if you want the full rafter length.
- What is the ridge length of a hip roof?
- The difference between the two plan dimensions: longer side minus shorter side. A 40 × 30 ft building has a 10 ft ridge, and a 36 × 24 ft building has a 12 ft ridge. The ridge always runs along the longer side, because all four planes rise at the same pitch to meet it.
- Why is the run half the shorter side rather than half the width?
- Because all four planes of an equal-pitch hip roof rise to a single ridge, the run is set by the shorter dimension. Using half the width unconditionally fails whenever the width exceeds the length: for a 30 × 40 ft plan at 6/12 it would give a 22.36 ft common rafter instead of the correct 16.77 ft. Swapping the two dimensions never changes the geometry.
- How do you calculate hip roof area?
- Multiply the footprint area, including any overhang, by the pitch factor. A hip roof covers its plan exactly once, so a 42 × 32 ft footprint at 6/12 gives 1,344 × 1.118034 = 1,502.6 sq ft. This is exact, not an approximation — decomposing the roof into two trapezoids and two triangles gives the identical figure.
- Why is the hip rafter shallower than the common rafter?
- Because it travels further horizontally for the same rise. A hip covers 16.97 inches of run for every 12 inches a common covers — the carpenter's "17 inch" rule. At 6/12 the common sits at 26.565° while the hip sits at 19.471°, so a hip rafter is longer but less steep.
- Is the hip overhang the same as the common rafter overhang?
- No, it is longer, because the hip extends diagonally across the corner. A 1 ft eave projection adds 1.118 ft along a common rafter at 6/12 but 1.500 ft along the hip. The ratio is √(2 + m²) ÷ √(1 + m²) where m is the rise over run, which is always above 1. Ordering hip stock to the common overhang length leaves it short.
- What happens when the building is square?
- The ridge length becomes zero and the roof is a pyramid: four triangular planes meeting at a single peak. There are no common rafters along a ridge, only the four hips and their jack rafters. The area formula still holds, since footprint × pitch factor equals the sum of the four triangles.
- How do you work out jack rafter lengths?
- Each jack is shorter than the last by a constant amount — the common difference, equal to the rafter spacing × pitch factor. At 16 in centres on a 6/12 that is 1.4907 ft per step. Each of the four hips carries jacks on both sides, so a hip roof has eight jack runs. Jacks also need a 45° side cut in plan where they meet the hip.
- What is the difference between a plumb cut and a seat cut?
- The plumb cut is the vertical cut where the rafter meets the ridge; the seat or level cut is the horizontal face of the birdsmouth that bears on the top plate. They are complementary. At 6/12 the plumb cut is 26.565° from vertical, which is the same angle as 63.435° measured from a line square across the rafter — both references are used on site, so check which one your square or saw is set to.