Sheet metal
Flat pattern cone calculator
In short
The flat pattern of a right frustum is a sector of a ring. With the half apex angle α (tan α = (Dg − Dp) / 2h), the large radius is R = (Dng / 2) / sin α, the small one r = (Dnp / 2) / sin α, and the sector angle θ = 360° × sin α, with diameters taken on the neutral fibre.
Calculation
Example calculated with the tool’s default values.
Inputs
- Large diameter
- 400 mm
- Small diameter (0 = full cone)
- 200 mm
- Height (between the two bases)
- 300 mm
- Plate thickness
- 3 mm
- The diameters entered are
- Outside
- Number of pieces (sector split)
- 1
- Generatrix step (on the base)
- 15°
Results
- Pattern radius (large arc)
- 627.5 mm
Worked example
Every value in this section belongs to the example, calculated with the tool’s default values.
Other results of the example
- Small arc radius
- 311.2 mm
- Sector angle
- 113.84°
- Large arc chord
- 1051.5 mm
- Large arc sagitta
- 285.0 mm
- Large arc length
- 1246.7 mm
- Small arc length
- 618.4 mm
- Generatrix (neutral fibre)
- 316.2 mm
- Half apex angle
- 18.43°
- Large Ø on neutral fibre
- 396.8 mm
- Small Ø on neutral fibre
- 196.8 mm
- Layout coordinates by generatrix
- X: along the chord of the large arc (0 at the middle); Z: distance from that chord towards the apex. One generatrix at each chosen step on the base.
n° base développé grand arc X ; Z petit arc X ; Z 0 0.0° -56.92° -525.8 ; 0.0 -260.8 ; -172.6 1 15.0° -52.18° -495.6 ; 42.3 -245.8 ; -151.6 2 30.0° -47.43° -462.1 ; 82.0 -229.2 ; -131.9 3 45.0° -42.69° -425.4 ; 118.7 -211.0 ; -113.7 4 60.0° -37.95° -385.8 ; 152.3 -191.4 ; -97.0 5 75.0° -33.20° -343.6 ; 182.5 -170.4 ; -82.0 6 90.0° -28.46° -299.0 ; 209.2 -148.3 ; -68.8 7 105.0° -23.72° -252.4 ; 232.0 -125.2 ; -57.5 8 120.0° -18.97° -204.0 ; 250.9 -101.2 ; -48.1 9 135.0° -14.23° -154.2 ; 265.7 -76.5 ; -40.8 10 150.0° -9.49° -103.4 ; 276.4 -51.3 ; -35.5 11 165.0° -4.74° -51.9 ; 282.8 -25.7 ; -32.3 12 180.0° 0.00° 0.0 ; 285.0 0.0 ; -31.2 13 195.0° 4.74° 51.9 ; 282.8 25.7 ; -32.3 14 210.0° 9.49° 103.4 ; 276.4 51.3 ; -35.5 15 225.0° 14.23° 154.2 ; 265.7 76.5 ; -40.8 16 240.0° 18.97° 204.0 ; 250.9 101.2 ; -48.1 17 255.0° 23.72° 252.4 ; 232.0 125.2 ; -57.5 18 270.0° 28.46° 299.0 ; 209.2 148.3 ; -68.8 19 285.0° 33.20° 343.6 ; 182.5 170.4 ; -82.0 20 300.0° 37.95° 385.8 ; 152.3 191.4 ; -97.0 21 315.0° 42.69° 425.4 ; 118.7 211.0 ; -113.7 22 330.0° 47.43° 462.1 ; 82.0 229.2 ; -131.9 23 345.0° 52.18° 495.6 ; 42.3 245.8 ; -151.6 24 360.0° 56.92° 525.8 ; 0.0 260.8 ; -172.6
- The pattern is a sector of radius R: add cutting allowance and, if needed, root gap on the straight edges. Indicative result.
Step-by-step calculation
Half-angle and generatrix
tan α = (Dg − Dp) / 2h ; s = √(h² + ((Dg − Dp)/2)²)
tan α = 100.0 / 300.0 → α = 18.43° ; s = 316.2 mm
Neutral-fibre diameters
Dn = D − e / cos α
e / cos α = 3.0 / 0.9487 = 3.16 → Dng = 396.8 ; Dnp = 196.8 mm
Pattern radii
R = (Dng / 2) / sin α ; r = (Dnp / 2) / sin α
R = 198.4 / 0.3162 = 627.5 mm ; r = 98.4 / 0.3162 = 311.2 mm
Sector angle
θ = 360° × sin α (arc = π × Dng)
θ = 360 × 0.3162 = 113.84°
Chord and sagitta
c = 2R sin(θ/2) ; f = R (1 − cos(θ/2))
c = 1051.5 mm ; f = 285.0 mm
Generatrix coordinates
φ = β sin α − θ/2 ; X = R sin φ ; Z = R cos φ − R cos(θ/2)
25 generators, pitch 15° on the base
Generatrices (sector layout)
| No. | Angle on the base (°) | Angle on the pattern (°) | Large arc X (mm) | Large arc Z (mm) | Small arc X (mm) | Small arc Z (mm) |
|---|---|---|---|---|---|---|
| 0 | 0.0 | −56.92 | −525.8 | 0.0 | −260.8 | −172.6 |
| 1 | 15.0 | −52.18 | −495.6 | 42.3 | −245.8 | −151.6 |
| 2 | 30.0 | −47.43 | −462.1 | 82.0 | −229.2 | −131.9 |
| 3 | 45.0 | −42.69 | −425.4 | 118.7 | −211.0 | −113.7 |
| 4 | 60.0 | −37.95 | −385.8 | 152.3 | −191.4 | −97.0 |
| 5 | 75.0 | −33.20 | −343.6 | 182.5 | −170.4 | −82.0 |
| 6 | 90.0 | −28.46 | −299.0 | 209.2 | −148.3 | −68.8 |
| 7 | 105.0 | −23.72 | −252.4 | 232.0 | −125.2 | −57.5 |
| 8 | 120.0 | −18.97 | −204.0 | 250.9 | −101.2 | −48.1 |
| 9 | 135.0 | −14.23 | −154.2 | 265.7 | −76.5 | −40.8 |
| 10 | 150.0 | −9.49 | −103.4 | 276.4 | −51.3 | −35.5 |
| 11 | 165.0 | −4.74 | −51.9 | 282.8 | −25.7 | −32.3 |
| 12 | 180.0 | 0.00 | 0.0 | 285.0 | 0.0 | −31.2 |
| 13 | 195.0 | 4.74 | 51.9 | 282.8 | 25.7 | −32.3 |
| 14 | 210.0 | 9.49 | 103.4 | 276.4 | 51.3 | −35.5 |
| 15 | 225.0 | 14.23 | 154.2 | 265.7 | 76.5 | −40.8 |
| 16 | 240.0 | 18.97 | 204.0 | 250.9 | 101.2 | −48.1 |
| 17 | 255.0 | 23.72 | 252.4 | 232.0 | 125.2 | −57.5 |
| 18 | 270.0 | 28.46 | 299.0 | 209.2 | 148.3 | −68.8 |
| 19 | 285.0 | 33.20 | 343.6 | 182.5 | 170.4 | −82.0 |
| 20 | 300.0 | 37.95 | 385.8 | 152.3 | 191.4 | −97.0 |
| 21 | 315.0 | 42.69 | 425.4 | 118.7 | 211.0 | −113.7 |
| 22 | 330.0 | 47.43 | 462.1 | 82.0 | 229.2 | −131.9 |
| 23 | 345.0 | 52.18 | 495.6 | 42.3 | 245.8 | −151.6 |
| 24 | 360.0 | 56.92 | 525.8 | 0.0 | 260.8 | −172.6 |
How to use it
- Enter the large diameter, the small diameter (0 for a full cone) and the height between the two bases.
- Enter the plate thickness and say whether the diameters are outside, inside or on the neutral fibre.
- Set the number of pieces if the sector is split, and the generatrix step for layout by coordinates.
- Read the pattern radius (large arc).
Frequently asked questions
What is the cone flat pattern formula?
Half apex angle: tan α = (Dg − Dp) / 2h; generatrix s = √(h² + ((Dg − Dp)/2)²). Pattern radii: R = (Dng / 2) / sin α and r = (Dnp / 2) / sin α = R − s. Sector angle: θ = 360° × sin α, because the sector arc equals the base circumference, π × Dng.
How do I work out a frustum flat pattern?
Example: Dg = 400 mm, Dp = 200 mm, h = 300 mm, 3 mm plate, outside diameters. tan α = 100.0 / 300.0, so α = 18.43° and s = 316.2 mm. On the neutral fibre, Dng = 396.8 and Dnp = 196.8 mm. R = 198.4 / 0.3162 = 627.5 mm, r = 311.2 mm, θ = 113.84°, large arc chord 1051.5 mm and sagitta 285.0 mm.
How do I develop a full cone?
Enter a small diameter of 0: the pattern becomes a disc sector of radius R = (Dng / 2) / sin α, with no small arc. The sector angle stays θ = 360° × sin α.
Should the plate thickness be taken into account?
Yes: the pattern is calculated on the neutral fibre. From an outside diameter, Dn = D − e / cos α; from an inside diameter, Dn = D + e / cos α. The e / 2 offset is normal to the plate: on a very open cone, e / cos α is clearly more than e.
How do I lay out a cone pattern too large for the plate?
The sector can be split into equal pieces, up to 12, welded afterwards. Beyond 180°, the chord alone no longer lays out the sector: the calculation also gives coordinates by generatrix (X along the large arc chord, Z towards the apex), useful to lay it out on the floor.
Limits
Indicative results. Right circular cone, bases cut square to the axis, dimensions taken on the neutral fibre. Add cutting allowance and, if needed, root gap on the straight edges. On plate that is thick relative to the diameter, rolling or hot forming can shift the neutral fibre: make a trial piece.
Sources
- Développement d’un cône droit : secteur de rayon s, angle 360° × r / s : Géométrie descriptive, développement de la surface latérale d’un cône droit (link)
- Calcul sur la fibre neutre (diamètre moyen = intérieur + e = extérieur − e) : Formation chaudronnier, module 2 chapitre 2 (cylindre, cône, piquage) (link)
- Trigonométrie du triangle rectangle : Décalage de fibre neutre normal à la tôle : e / cos α sur le diamètre (géométrie)