If a sphere is not developable, what does this pattern guarantee?

The true length of the meridian, down the middle of the gore, and the mouth: each station is the arc of its parallel developed on the cone tangent to the sphere at that height, with the half-width measured along the arc, so the mouths of the gores add up to the full round and the mouth corner comes out square. The seams stay close to the meridian — 1.85% longer in the example, against 5.7% with straight stations. The AREA does not close exactly: it is 0.9% over in the example, and the assembly note gives the figure for each size. That is what forming absorbs.

The more the better — in the example, with 6 gores the seam runs 3.3% past the meridian and the area is 1.6% over; with 8, 1.85% and 0.9%; with 12, 0.8% and 0.4%. Eight to twelve is usual. Below three the flat sheet would have to stretch a large fraction of its own length, and the calculation refuses.

Can the rise exceed half the diameter?

No. At half the diameter the head is a hemisphere, and past that the mouth stops being the widest parallel: the gore starts to belly, and a layout by meridian and parallel no longer describes the part.

And the sphere's radius, where does that come from?

From the mouth and the rise, which is what you have in hand. It is (d²/4 + h²) over 2h. The calculation does the arithmetic; you give the two dimensions of the head.

Sphere Gore (Dished Head Petal) flat pattern

What it is

One gore of a spherical head, and as many as the head takes. It is NOT an exact development, and the calculation says so: a sphere is not developable, no flat sheet becomes a cap without stretching. What comes out here is the layout from the plate-work books — true length along the meridian, and each station on the arc of its developed parallel, with the half-width measured along the arc — and the gore is formed afterwards, in a press or hot.

Where it is used

Tank and silo heads, vessel caps, reservoir domes, spherical equipment bottoms.

Measurements the calculation needs

D Diameter of the opening
H Rise (height of the head)
S Number of gores
DV Divisions
THK Plate thickness

Calculate with your own measurements

No account needed: enter the measurements in millimetres or inches and get the figures and the drawing. The scaled PDF template, the DXF for the cutting table and the 3D model take a free account.

Diameters and sides are outside measurements. The flat pattern already allows for the plate thickness, working on the mean line.

Units: mm inches

How it develops: the formulas

Sphere radius: R = (D²/4 + h²) / (2 × h) D is the mouth diameter and h the rise. For D = 2000 and h = 500: R = 1250 mm.
Meridian angle and length: φ0 = arcsin( D / (2R) ) ; meridian = R × φ0 φ0 = 53.13° = 0.927 rad. On the outside face the meridian measures 1250 × 0.927 = 1159 mm; the template measures it on the mean surface, 1245 × 0.927 = 1154.5 mm, and that is the length of the gore in the example.
Half width of the gore at each station, measured along the arc: w(φ) = π × R × sin φ / s s is the number of gores, φ the angle from the pole and R the radius on the mean surface, R − t/2. At the mouth, for 8 gores in 10 mm plate: w = π × 1245 × 0.8 / 8 = 391.1 mm of half width, measured along the mouth arc.
Arc of each station: ρ(φ) = R × tan φ ; sweep = π × cos φ / s Each station is struck with radius ρ from a centre on the middle line of the gore, on the pole side: the apex of the cone tangent to the sphere at that station. The half width w takes up the sweep on that arc. At the mouth in the example: ρ = 1245 × tan 53.13° = 1660 mm and a sweep of 13.5° each side, 27° over the whole arc, giving a 775.0 mm chord and a 45.9 mm rise.

Worked example

The values the form comes pre-filled with, run through the tool itself:

Diameter of the opening	2000 mm
Rise (height of the head)	500 mm
Number of gores	8
Divisions	24
Plate thickness	10 mm
The flat pattern fits a plate of	1154 × 775 mm

Result of the example

Plate	1154 × 775 mm, cut 8 times
Area and weight	0.49 m² · ≈ 38.6 kg in 10 mm steel
Arc radii	1,660 mm
Sector angle	27°

Sphere Gore (Dished Head Petal) flat pattern: drawing of the template with the worked example dimensions — Cut lines in purple, welds in green and reference lines in orange. Plate 1154 × 775 mm.

Assembly notes

A sphere is NOT developable: no flat sheet becomes a cap without stretching. This pattern is the boilermaker's layout from the plate-work books — true length along the meridian, and each station on the arc of its developed parallel, with the half-width measured along the arc — and the gore has to be formed, in a press or hot. The seam comes out close to the meridian's length and the mouth corner square; the AREA is 0.9% over at these dimensions, and that surplus is what forming absorbs. More gores, less surplus.

Weld the sheets with the longitudinal seams 180° apart, not lined up. Lined up, the piece assembles into a different shape — and the longitudinal seam crosses the circumferential one.

Guides about this part

Frequently asked questions

If a sphere is not developable, what does this pattern guarantee?: The true length of the meridian, down the middle of the gore, and the mouth: each station is the arc of its parallel developed on the cone tangent to the sphere at that height, with the half-width measured along the arc, so the mouths of the gores add up to the full round and the mouth corner comes out square. The seams stay close to the meridian — 1.85% longer in the example, against 5.7% with straight stations. The AREA does not close exactly: it is 0.9% over in the example, and the assembly note gives the figure for each size. That is what forming absorbs.
How many gores?: The more the better — in the example, with 6 gores the seam runs 3.3% past the meridian and the area is 1.6% over; with 8, 1.85% and 0.9%; with 12, 0.8% and 0.4%. Eight to twelve is usual. Below three the flat sheet would have to stretch a large fraction of its own length, and the calculation refuses.
Can the rise exceed half the diameter?: No. At half the diameter the head is a hemisphere, and past that the mouth stops being the widest parallel: the gore starts to belly, and a layout by meridian and parallel no longer describes the part.
And the sphere's radius, where does that come from?: From the mouth and the rise, which is what you have in hand. It is (d²/4 + h²) over 2h. The calculation does the arithmetic; you give the two dimensions of the head.

Sphere Gore (Dished Head Petal) dimension drawing: D (Diameter of the opening), H (Rise (height of the head)), S (Number of gores), DV (Divisions), THK (Plate thickness), marked on the part

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