Cone Flat Pattern Calculator
Find the sector angle, slant height, arc lengths, and pattern area needed to cut a flat sheet that rolls into a cone or frustum (truncated cone). Used for sheet metal fabrication, craft, and geometry.
Flat Pattern Calculator
Presets:
Flat Pattern Shape
Step-by-Step Working
Common Cone Patterns Reference
| Type | Cone Shape | Sector Angle | Pattern Radius | Notes |
|---|---|---|---|---|
| Flat (disc) | H = 0 | 360° | L = R | Degenerate — full circle, no height |
| Shallow | R ≈ H | ~254° | L ≈ 1.41 R | Wide funnel shape |
| Right angle | H = R | ≈ 254.6° | L = R√2 | 45° half-angle at apex |
| Party hat | H ≈ 2R | ~160° | L ≈ 2.24 R | Typical party/dunce hat proportions |
| Narrow | H = 5R | ≈ 71° | L ≈ 5.1 R | Tall spike/pointy ice cream cone |
| Traffic cone | R=15, H=52 cm | ≈ 99° | L ≈ 54 cm | Standard 52 cm road cone |
How to Create a Cone Flat Pattern
A cone flat pattern (also called cone development or cone layout) is the 2D shape that, when cut and rolled, forms a 3D cone. The flat pattern is a sector of a circle — a pie-slice shape with a straight cut from the center to the edge.
Full Cone Formula
For a cone with base radius R and vertical height H:
- Slant height: L = √(R² + H²)
- Sector angle: θ = (R ÷ L) × 360°
- Arc length: 2πR (equals the circumference of the cone base)
- Pattern area: π × R × L (lateral surface area)
The flat pattern is a sector of radius L. Cut the sector, then roll the two straight edges together and they form the cone's slant side meeting at the apex.
Frustum (Truncated Cone) Formula
A frustum is a cone with the tip cut off. With bottom radius R, top radius r, and height H:
- Frustum slant length: Lf = √(H² + (R − r)²)
- Outer pattern radius: L_big = R × Lf ÷ (R − r)
- Inner pattern radius: L_small = r × Lf ÷ (R − r)
- Sector angle: θ = (R ÷ L_big) × 360°
- Lateral pattern area: π × Lf × (R + r)
The frustum flat pattern is an annular sector — a ring-slice between two arcs of radii L_big and L_small with sector angle θ.
Applications
- Sheet metal fabrication: hoppers, funnels, reducers, silos, exhaust cones
- Crafts and paper: party hats, gift cones, origami models
- Woodworking: turned wooden cones, lamp shades
- Industrial: HVAC duct transitions, pipe reducers, nozzle patterns
Frequently Asked Questions
A cone flat pattern (or cone development) is the 2D shape you cut from flat sheet material that, when rolled up, forms a cone. It is a sector of a circle — a pie-slice shape — where the radius equals the slant height of the cone and the arc length equals the circumference of the cone's base. The sector angle θ = (R/L) × 360°, where R is the base radius and L is the slant height.
The sector angle θ (in degrees) = (Base Radius R ÷ Slant Height L) × 360°. The slant height L = √(R² + H²) where H is the cone height. For example, a cone with R = 5 cm and H = 12 cm has L = √(25 + 144) = 13 cm, and θ = (5/13) × 360° = 138.5°.
For a frustum with bottom radius R, top radius r, and height H: 1) Frustum slant Lf = √(H² + (R−r)²). 2) Outer radius L_big = R × Lf ÷ (R−r). 3) Inner radius L_small = r × Lf ÷ (R−r). 4) Sector angle θ = (R / L_big) × 360°. The flat pattern is an annular sector (ring sector) between arcs of radii L_big and L_small.
The flat pattern area equals the lateral surface area of the cone: A = π × R × L, where R is the base radius and L is the slant height. For a frustum, the lateral area = π × Lf × (R + r). This is the amount of flat sheet material needed (excluding the circular base(s)).
This calculator gives the theoretical flat pattern dimensions with no seam allowance. For sheet metal, add 3–10 mm to the straight edges of the sector for a weld or rivet seam. For paper/card, overlap of 10–15 mm on the arc edge is usually sufficient. For fabric, add your standard seam allowance (typically 12 mm or ½ inch) along the straight edges of the sector.