Cone Frustum Dimensions

Cone & Frustum Pattern Calculator

Calculation Results & Developed Flat Patterns

Full-scale developed annular sector flat pattern blueprint, frustum cross-section elevation, and station ordinates:

📐 Truncated Cone — Developed Flat Annular Sector Pattern
Cone Frustum Flat Pattern Blueprint Scaled flat annular sector pattern showing outer arc Ls, inner arc Lt, radial seam lines, and swept angle θ.

Print template at 100% scale without page scaling — or scribe calculated outer/inner radii directly onto sheet metal stock.

⭕ Truncated Cone Side Elevation Geometry

Truncated Cone Frustum Elevation Side elevation diagram showing base diameter D, top diameter d, vertical height h, and slant height L.
Outer Slant Radius (Ls)
17.088 in
Apex to base distance
Inner Slant Radius (Lt)
8.544 in
Apex to top distance
Included Sector Angle (θ)
126.40°
θ = 360° · R / Ls
Outer Arc Length
37.699 in
✓ Circumference = π · D

🔨 Outer Arc Station Marking Table

Sequential trammel/compass arc mark points along outer Developed Arc:

Station # Sub-Angle (θi) Arc Distance around Base Straight Chord from Start

📦 Required Material & Sheet Metal Checklist

Select Item Description Est. Quantity Fabrication Notes

Related Development Flat-Pattern Templates

⚠️ Sheet Metal Fabrication Disclaimer: Calculations and templates are provided for layout planning. Always wear protective leather gloves and eye protection when cutting or handling sheared sheet metal edges. Verify SMACNA duct gauge standards before installing HVAC transitions.

🧮 Step-by-Step Worked Example: Truncated Cone Reducer Layout

Consider laying out a truncated cone frustum reducer with a 12.000 inch base diameter (D) (R = 6.000"), a 6.000 inch top diameter (d) (r = 3.000"), and a 8.000 inch vertical height (h) with a 0.500 inch seam lap.

Mathematical Derivation Steps:

  1. Base & Top Radii Difference (ΔR):
    ΔR = R - r = 6.000 in - 3.000 in = 3.000 inches (76.20 mm)
  2. Full Cone Apex Height (H):
    H = h × R / ΔR = 8.000 in × 6.000 / 3.000 = 16.000 inches (406.40 mm)
  3. Outer Slant Radius (L_s):
    L_s = √(H² + R²) = √(16.000² + 6.000²) = √(256 + 36) = √292 = 17.088 inches (434.04 mm)
  4. Inner Slant Radius (L_t):
    L_t = L_s × (r / R) = 17.088 in × (3.000 / 6.000) = 8.544 inches (217.02 mm)
  5. Included Sector Angle (θ):
    θ = 360° × (R / L_s) = 360° × (6.000 / 17.088) = 126.40°
  6. Outer Arc Length (L_arc):
    L_arc = π × D = 3.14159 × 12.000 in = 37.699 inches (957.56 mm)

🛠️ Sheet Metal Equipment, Safety (PPE) & Cone Layout Pitfalls

Layout & Forming Tools

  • Beam compass & trammel points
  • Aviation hand snips (Red/Green/Yellow) & Beverly shear
  • 3-Roll initial-pinch slip roll machine
  • Tungsten carbide scribe & steel rule
  • Spot welder, pop rivet gun, or soldering iron

Required Personal Safety (PPE)

  • Heavy leather work gloves (ANSI Cut Level 4 rating)
  • ANSI Z87.1 approved impact safety glasses
  • Flame-resistant shop shirt / leather sleeves
  • Ear protection during motorized shearing
  • Steel-toe leather work boots (ASTM F2413)

Critical Sheet Metal Layout Pitfalls

  1. Forgetting Seam Lap Allowance: Cutting the developed sector without adding seam lap width results in an undersized base diameter after joint overlap.
  2. Scribing Lines with Soft Pencil: Marking arcs with thick pencil lead introduces up to 1/16" error; always use sharp trammel points or a tungsten scribe.
  3. Lap Allowance Overlapped on Wrong Side: Folding seam tabs inward instead of outward changes internal cone dimensions and restricts airflow.
  4. Flat Seam Edges from Slip Rolling: Rolling sheet metal without pre-creasing the leading 1" seam edges creates flat spots along the longitudinal joint.

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Frequently Asked Questions (FAQ) — Cone & Frustum Pattern Development

What is a cone frustum in sheet metal development?

A frustum is a truncated cone formed by slicing off the apex top parallel to the base, producing a conical reducer transition with different top and bottom diameters.

How is the apex height H of a full cone calculated?

Apex height H from base to apex is calculated as H = h × R / (R - r), where h is vertical frustum height, R is base radius (D/2), and r is top radius (d/2).

How is outer slant radius L_s calculated?

Outer slant radius L_s is the true length from apex to outer base edge: L_s = √(H² + R²), forming the outer radius of the developed flat pattern sector.

How is inner slant radius L_t calculated?

Inner slant radius L_t is the distance from apex to top cut edge: L_t = L_s × (r / R), forming the inner radius arc of the developed annular sector.

What is the included sector angle θ formula?

The included sector angle θ = 360° × (Base Radius R / Slant Radius L_s), representing the total arc angle swept by the flat developed pattern.

How is seam lap allowance added to the flat cone layout?

Seam lap width (e.g. 0.5") is added as a parallel tab along one radial seam edge for spot welding, riveting, or lap soldering.

What is the difference between concentric and eccentric cone reducers?

Concentric cones have aligned top and bottom centerlines. Eccentric cones have one straight flat side, requiring asymmetric triangulation pattern development.

How do SMACNA HVAC standards govern sheet metal cone reducers?

SMACNA HVAC Duct Construction Standards specify sheet metal gauge (e.g. 24 Ga to 16 Ga galvanized steel) and seam types (Pittsburgh lock or standing seam) based on duct diameter and pressure.

What tool is used to scribe large cone arc radii on sheet metal?

Trammel points attached to a steel beam compass (or a tight wire pivot anchor) are used to scribe precision arcs up to several feet in radius.

How are flat cone patterns formed into round 3D cones?

Cut sheet metal sector patterns are passed through 3-roll initial-pinch slip rollers, adjusting roll tilt to match the conical taper.

What causes flat spots near radial seams when rolling sheet metal cones?

Slip rolls cannot bend the leading and trailing 1 to 2 inches of sheet stock unless the seam edges are pre-bent using hand brake tongs or forming dies.

How do 1:1 printable paper templates aid cone fabrication?

Printing a 1:1 sector template allows fabricators to paste the paper directly onto raw sheet stock and cut along outer and inner arcs with hand snips.

What station spacing is recommended for manual arc marking?

12 stations (30° base increments) provide accurate layout chord points for small duct transitions; 24 stations are used for large hopper cones.

How is neutral axis offset calculated for thick plate cone rolling?

For metal plate over 1/8" (3.2 mm) thick, calculations must use mean diameter (OD - thickness) to account for neutral axis stretch during plate bending.

What joint preparation is required for butt-welding heavy plate cones?

Radial seam edges must be beveled at 30° to 37.5° for V-groove welds with 100% root penetration per AWS D1.1 structural welding code.

How do you calculate a sharp-pointed full cone pattern?

Set top diameter d = 0. The inner slant radius L_t becomes 0, producing a solid circular sector without a top hole cut-out.

What sheet metal snips are best for cutting curved annular sector arcs?

Aviation snips (Red for left-curving arcs, Green for right-curving arcs, Yellow for straight seam lines) or a power throatless Beverly shear.

What personal protective equipment is mandatory when handling sheet metal?

Cut-resistant leather gloves (ANSI Cut Level 4+), safety glasses, long sleeves, and steel-toe boots are required to prevent severe lacerations from raw sheet edges.

Sources & Sheet Metal Standards

  1. AWS Structural & Sheet Metal Welding Standard: AWS D1.3 Structural Welding Code — Sheet Steel View Standard
  2. WBDG UFGS 07 62 00 Sheet Metal Flashing Specification: WBDG UFGS Specification View Standard
  3. Industrial Press Machinery's Handbook: Truncated Cone Sector Layout View Reference