Cone Flat Pattern Template Generator

What is a Cone Flat Pattern?

A cone flat pattern represents the flat 2D shape that must be cut out of a flat sheet of material (such as paper, cardboard, or sheet metal) so that when rolled up, it forms a 3D cone or a truncated cone (frustum). This process of transforming a 3D shape into a 2D flat layout template is widely used in metal fabrication, HVAC ductwork design, boiler making, and various DIY crafting projects.

Flat Layout Math for a Full Cone

For a complete cone with a base radius $R$ and vertical height $h$, the flat pattern is a simple circular sector. The math behind the pattern is:

1. Slant Height (Pattern Radius $L$)

$$L = \sqrt{R^2 + h^2}$$

2. Sector Angle ($\theta$)

$$\theta = 360 \times \frac{R}{L}$$

The arc length of this sector is equal to the circumference of the cone base: $2 \pi R$. You draw a circle of radius $L$, mark a sector of angle $\theta$, and cut it out to form the cone.

Flat Layout Math for a Truncated Cone (Frustum)

A truncated cone has a top radius $r$, bottom radius $R$, and vertical height $h$. The flat pattern is a sector of an annulus (ring sector), defined by the following formulas:

1. Slant Height ($L$)

$$L = \sqrt{(R - r)^2 + h^2}$$

2. Outer Pattern Radius ($R_{flat}$)

$$R_{flat} = \frac{R \times L}{R - r}$$

3. Inner Pattern Radius ($r_{flat}$)

$$r_{flat} = R_{flat} - L$$

4. Sector Angle ($\theta$)

$$\theta = 360 \times \frac{R - r}{L}$$

To draw this, scribe two concentric arcs with radii $R_{flat}$ and $r_{flat}$ using the center point, measure the sector angle $\theta$, and cut the resulting shape. For related geometric tools, try our Cone Calculator, Conical Frustum Calculator, or Arc Length Calculator.

Frequently Asked Questions

What is the difference between a full cone and a truncated cone?

A full cone tapers to a single point at the top. A truncated cone (also called a frustum of a cone) has its top cut off parallel to the base, resulting in two flat circular openings of different diameters.

How do I transfer the flat pattern to a piece of sheet metal?

Use the calculated values of Outer Radius ($R_{flat}$) and Inner Radius ($r_{flat}$) to draw two concentric circles from a single center point using a compass. Then use a protractor to mark the sector angle ($\theta$) and cut along the lines.

Why must the top diameter be smaller than the bottom diameter?

By definition, a frustum tapers from a larger base to a smaller top. If the diameters were equal, it would be a cylinder, which requires a simple rectangular flat layout rather than an annular sector.

Does this calculator account for material thickness or seams?

No, this calculator computes the theoretical layout of the median line. For real-world sheet metal fabrication, you must add extra material margins for weld or rivet seams, and adjust for the neutral bending line if using thick materials.

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