Pyramid Hip Roof Calculator
About Pyramid Hip Roof Calculation
The Pyramid Hip Roof Calculator performs a geometric roofing calculation for a rectangular building with a single apex and four roof slopes. It calculates the roof dimensions at the eaves, slope angles, roof area, rafter system, battens, and selected additional materials for designing pyramid hip roofs and other four-sided roofing structures.
All input linear dimensions are entered in millimetres. Material lengths are shown in metres, areas in m2, and volumes in m3. Roof overhangs are treated as extensions of the same roof planes with unchanged slope angles.
Guidelines and Recommendations
Pyramid Hip Roof Geometry
Roof dimensions. Let the building length be A, the building width be B, the roof height above the wall level be H, and the eaves overhang be C. The overhang is added to both sides of each building dimension:
Ar = A + 2 × C
Br = B + 2 × C
Roof slope angles. On a rectangular pyramid hip roof, the two pairs of slopes have different horizontal runs. The slope with eaves length A rises to the apex over half of width B, while the slope with eaves length B rises over half of length A:
αA = arctan(H / (B / 2))
αB = arctan(H / (A / 2))
The angles are displayed to the nearest 0.1°. For a square building plan where A = B, both roof slope angles are equal.
Roof Area
Slope factors. To convert the horizontal projection into the actual sloping roof area, the calculator determines factors equal to the ratio between the inclined length and its horizontal projection:
kA = √((B / 2)2 + H2) / (B / 2)
kB = √((A / 2)2 + H2) / (A / 2)
Area of the four roof slopes. The two pairs of triangular slopes are calculated separately using their corresponding slope factors, then their areas are added together:
S = Ar × Br × (kA + kB) / (2 × 1000000)
The factor 1000000 converts mm2 to m2. The resulting roof area is rounded upward to 0.1 m2 so that rounding does not reduce the calculated roofing quantity.
Rafter Layout
Specified spacing. Rafter spacing is treated as the nominal distance between neighbouring rafters. The rafters are arranged symmetrically relative to the roof axes. If a partial interval remains near the edge of a slope, the outer rafters are adjusted to avoid leaving a very narrow residual gap.
Even distribution. When a maximum rafter spacing is specified, the calculator first determines the available distance while accounting for the thickness of the diagonal rafters. For the roof aspect ratio r = Ar / Br, the projected components of rafter thickness S2 are:
pB = S2 / √(1 + r2)
pA = r × pB
The larger available distance from the two directions is then selected:
Lavail = max(Ar - 2pA - S2, Br - 2pB - S2)
The number of intervals is rounded upward so that the specified maximum spacing is not exceeded:
n = ceil(Lavail / Pmax)
P = floor(Lavail / n)
The actual spacing P is rounded downward to the nearest whole millimetre and then used for the symmetrical rafter layout.
Rafter Length and Volume
Diagonal rafters. The calculator uses 4 diagonal rafters running from the four roof corners to the apex. First, the horizontal diagonal projection within the building outline is calculated:
G = √((A / 2)2 + (B / 2)2)
A spatial slope factor is then calculated for this projection:
kd = √(G2 + H2) / G
The diagonal projection to the outer eaves corners is:
Gr = √((Ar / 2)2 + (Br / 2)2)
The calculated length of one diagonal rafter includes an allowance for the lower angled cut of a board with width S1:
Ld = ceil(Gr × kd + S1 × √(kd2 - 1))
Intermediate rafters. For each rafter position, its horizontal length to the apex is determined. This length is multiplied by the slope factor of the corresponding roof face, and the allowance for the lower angled cut is then added:
Li = ceil(Xi × k + S1 × √(k2 - 1))
Here, Xi is the horizontal projection of the individual rafter and k is the slope factor for that roof face. Individual rafter lengths are rounded upward to the nearest 1 mm.
Total rafter volume. The lengths of all intermediate rafters and the four diagonal rafters are added together. The timber volume is calculated from the rafter cross-section:
Vraf = Lraf × S1 × S2 / 1000000000
In this formula, Lraf is expressed in millimetres. Division by 1000000000 converts mm3 to m3.
Battens
Row layout. The first batten row is positioned at the eaves. If no separate first spacing is specified, the second row is positioned using the main batten spacing O3. If a separate first spacing is specified, only the distance between the first and second rows uses that value, while all subsequent intervals return to O3.
Last row. Rows are added toward the roof apex while accounting for batten width O1. If the remaining free distance after the last calculated row is greater than max(120 mm, O1), the calculator adds one more row near the upper part of the slope.
Row length. The length of each batten row is determined from the actual width of the triangular roof slope at its distance from the eaves. The row length decreases linearly toward the apex. The total batten length is obtained by adding the rows across all four roof slopes.
Batten volume. The unrounded total batten length in millimetres is used for the volume calculation:
Vbat = Lbat × O1 × O2 / 1000000000
The total batten length shown in the results is rounded to the nearest whole metre, while the volume is rounded to 0.01 m3.
Additional Timber Elements
Fascia board. Its length is taken as the complete outer roof perimeter along the eaves:
Lf = 2 × (Ar + Br)
Wall plate. The wall plate is calculated around the building perimeter without the roof overhangs. Four wall plate widths M are subtracted from the total perimeter to account for the four corners:
Lw = max(0, 2A + 2B - 4M)
Counter-battens. Their total length is taken as equal to the total rafter length because the counter-battens run along the rafter members:
Lcb = Lraf
Timber volume. The same principle is used for each timber element:
V = L × b × h / 1000000000
Here, L, b, and h are expressed in millimetres. Rafters and battens are always included in the total timber volume. The fascia board, wall plate, and counter-battens are added only when their calculation is enabled:
Vtotal = Vraf + Vbat + Vf + Vw + Vcb
Waterproofing
Material area. The calculated roof area S is used as the starting value. Additional material for overlaps is determined from the roll length Lr, roll width Wr, and overlap N:
Sw = S × (1 + N / Lr + N / Wr)
The ratios inside the brackets are dimensionless, so Lr, Wr, and N must use the same units. The calculated waterproofing area is rounded to 0.1 m2.
Number of rolls. The waterproofing area already rounded to 0.1 m2 is used for the next step:
Qr = Sw / (Lr × Wr / 1000000)
The number of rolls is displayed to the nearest 0.1 and is not automatically rounded up to a whole roll. The total calculated roll length is determined as Qr × Lr and displayed to the nearest 0.1 m.
Insulation
Insulated roof area. For insulation, the calculator uses the roof slope area above the building outline without the eaves overhangs. It is determined using the same slope factors:
Sins = A × B × (kA + kB) / (2 × 1000000)
Insulation volume. For an insulation thickness U in millimetres:
Vins = Sins × U / 1000
Division by 1000 converts the insulation thickness from millimetres to metres. The result is rounded to 0.01 m3.
Practical Guidelines and Assumptions
Rafter spacing. Timber pitched roofs often use rafter spacing of about 600 mm. Values around 600-800 mm are also common, but the actual spacing depends on the span, timber section, insulation dimensions, and design loads.
Roof overhang. Overhangs of approximately 300-700 mm are common for low-rise buildings. Increasing the overhang increases the roofing area, fascia board length, and batten quantity.
First batten spacing. For some metal roof tile systems with a main spacing of about 350 mm, a first spacing of approximately 280-300 mm is common. The exact value depends on the roof profile and the manufacturer's specified measuring method.
Waterproofing overlap. Many roll roofing membranes commonly use overlaps of about 100-150 mm. The required overlap depends on the membrane type, roof slope, and method used to join adjacent sheets.
Material allowance. The calculated lengths, areas, and volumes are geometric quantities. No additional purchasing waste factor is applied automatically. For timber, an additional allowance of about 5-10% is often used for cutting, grading, joints, and waste.
Related European Standards
EN 1990 Eurocode. Basis of structural design. Defines the general principles of structural reliability, design situations, and combinations of actions used when designing roof structural elements.
EN 1991-1-3 Eurocode 1. Actions on structures. Part 1-3: General actions. Snow loads. Used to determine snow actions while accounting for roof shape and slope, climate conditions, and roof shape coefficients.
EN 1991-1-4 Eurocode 1. Actions on structures. Part 1-4: General actions. Wind actions. Provides rules for determining wind pressure and suction, including the effects of roof shape, building height, and roof edge zones.
EN 1995-1-1 Eurocode 5. Design of timber structures. Part 1-1: General rules and rules for buildings. Used for selecting timber rafter sections and checking strength, deflection, stability, and connections.
FAQs
Why can a pyramid hip roof have two different slope angles?
With a rectangular building plan, the apex is located above the centre of the building, but the distances from the apex to eaves A and B are different. Therefore, with the same roof height H, the two pairs of roof slopes have different horizontal runs and different angles. With a square plan, both angles become equal.
How does the calculator choose the rafter spacing for even distribution?
It first determines the available distance while accounting for the roof geometry and the thickness of the diagonal rafters. This distance is then divided into the smallest whole number of intervals that keeps the actual spacing at or below the specified maximum. The resulting spacing is rounded downward to the nearest whole millimetre.
Why are diagonal rafters longer than the other rafters?
A diagonal rafter runs in two horizontal directions at the same time, from an eaves corner to the roof apex. Its horizontal projection is therefore longer than that of an intermediate rafter. The calculated length also includes an allowance for the lower angled cut of the board.
When is a separate first batten spacing required?
A separate first spacing is used for roofing systems where the distance between the first and second batten rows differs from the main spacing. After the second row, the calculator returns to the standard batten spacing. This arrangement is common with some metal roof tile profiles.
Why can the calculated waterproofing quantity be fractional?
The number of rolls represents the mathematical ratio between the required waterproofing area and the area of one roll, including the specified overlap. The result can therefore be, for example, 3.4 rolls. For purchasing, a fractional quantity is normally rounded upward while also considering an appropriate material allowance.