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

The AREA, exactly. The gore's area is the integral of the parallel divided by the number of gores, which comes to exactly one s-th of the cap's area — added up, the gores have the area of the part, with nothing over and nothing missing, so the material ordered is right. What is approximate is the SHAPE, and forming is what settles that.

The more the better — the shape error falls with the width of the gore. 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 shop layout — true length along the meridian, half-width from the parallel at each station — 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: 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.

Worked example

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

Diameter of the opening	78-3/4" (78.7402)
Rise (height of the head)	19-11/16" (19.685)
Number of gores	8
Divisions	24
Plate thickness	3/8" (0.3937)
The flat pattern fits a plate of	45-7/16" × 30-13/16"

Result of the example

Plate	45-7/16" × 30-13/16", cut 8 times
Area and weight	5.2 ft² · ≈ 84.3 lb in 3/8" steel

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 45-7/16" × 30-13/16".

Assembly notes

A sphere is NOT developable: no flat sheet becomes a cap without stretching. This pattern is the shop layout — true length along the meridian and half-width from the parallel at each station — and the gore has to be formed, in a press or hot. The AREA of the pattern is exact; the shape is approximate, and the approximation improves with more gores.

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 AREA, exactly. The gore's area is the integral of the parallel divided by the number of gores, which comes to exactly one s-th of the cap's area — added up, the gores have the area of the part, with nothing over and nothing missing, so the material ordered is right. What is approximate is the SHAPE, and forming is what settles that.
How many gores?: The more the better — the shape error falls with the width of the gore. 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

Calculate here, no account Open the full calculator

You can work out the figures right here, with no account. PDF, DXF, 3D and history take an account — free, with unlimited calculations.