Truncated Cone Pattern Calculator: Formula & Guide

This calculator computes the flat pattern dimensions needed to fabricate a truncated cone (conical frustum) from sheet metal, paper, or other flat materials. You enter the top diameter, bottom diameter, and height of the cone, and it returns the radii of the two arcs and the angle of the sector that form the developed pattern.

Formula

The pattern of a truncated cone is an annular sector—a portion of a ring. The two radii of the ring are the slant heights of the full cone and the cut-off tip. Given top diameter \(d_1\), bottom diameter \(d_2\), and perpendicular height \(h\), the slant height of the full cone (large radius \(R_1\)) is:

\(R_1 = \frac{d_2}{2} \sqrt{1 + \left(\frac{2h}{d_2 – d_1}\right)^2}\)

The small radius \(R_2\) (slant height of the removed tip) is proportional:

\(R_2 = \frac{d_1}{2} \sqrt{1 + \left(\frac{2h}{d_2 – d_1}\right)^2}\)

The sector angle \(\theta\) (in degrees) is the angle that, when the pattern is rolled, makes the bottom circumference match \(\pi d_2\):

\(\theta = \frac{d_2}{2 R_1} \times 180^\circ\)

All lengths are in the same unit (mm, cm, inches). The angle is independent of units.

Worked Example

Using the default values: top diameter \(d_1 = 100\,\text{mm}\), bottom diameter \(d_2 = 200\,\text{mm}\), height \(h = 150\,\text{mm}\).

1. Compute the difference: \(d_2 – d_1 = 100\,\text{mm}\).
2. Compute \(\frac{2h}{d_2-d_1} = \frac{300}{100} = 3\).
3. Square it: \(3^2 = 9\), add 1: \(10\).
4. Square root: \(\sqrt{10} \approx 3.1623\).
5. Large radius: \(R_1 = \frac{200}{2} \times 3.1623 = 100 \times 3.1623 = 316.23\,\text{mm}\).
6. Small radius: \(R_2 = \frac{100}{2} \times 3.1623 = 50 \times 3.1623 = 158.11\,\text{mm}\).
7. Sector angle: \(\theta = \frac{200}{2 \times 316.23} \times 180 = \frac{200}{632.46} \times 180 \approx 56.92^\circ\).

The calculator shows \(R_1 = 316.23\,\text{mm}\), \(R_2 = 158.11\,\text{mm}\), \(\theta = 56.92^\circ\). To draw the pattern, mark a center point, draw two arcs with those radii, and cut out the sector with that angle.

How to Use the Pattern

After obtaining the two radii and the sector angle, you can lay out the pattern on flat material:

• Draw a straight line from the center point; this is one straight edge of the sector.
• Using a protractor or compass, mark the sector angle from that line.
• Draw the second straight edge from the center at that angle.
• With the center as pivot, draw arcs of radius \(R_1\) and \(R_2\) between the two lines.
• Cut out the annular sector; the outer arc becomes the bottom rim, the inner arc the top rim.

When rolling, the straight edges join to form the seam. The overlap allowance for welding, riveting, or gluing is not included—add that separately to one straight edge.

Common Mistakes and Unit Mix-Ups

• Using radius instead of diameter: The formula expects diameters. If you measure radii, double them first.
• Mixing units: All three inputs must be in the same unit. Convert inches to mm (1 in = 25.4 mm) before entering.
• Confusing height with slant height: The height is the perpendicular distance between the two circular faces, not the length along the cone’s side.
• Forgetting the seam allowance: The pattern is the exact cone surface; any joint overlap must be added manually.
• Rounding too early: Keep at least four significant figures in intermediate steps to avoid noticeable errors in the angle.

Also note that the formula assumes a perfectly straight-sided cone. If the cone has a curved profile (e.g., a bell shape), this pattern will not be accurate.

When to Use This Calculator

This tool is for fabricating a frustum with flat, circular top and bottom faces. Typical uses include:

• Sheet metal transitions for ductwork, hoppers, or funnels.
• Paper or cardboard cones for crafts, party hats, or model making.
• Fabric patterns for lampshades or conical skirts.
• Any project where you need to cut a flat shape that rolls into a truncated cone.

If the cone is full (top diameter zero), you can still use this calculator by entering a very small top diameter, but the formula becomes singular at exactly zero. For a full cone, use the simpler formula: radius = slant height, angle = (base diameter / (2 × slant height)) × 180°.

Limitations of the Formula

The formula gives the exact developed surface for a mathematical frustum. Real-world factors to consider:

• Material thickness: For thick sheet metal, the neutral axis (where the material neither stretches nor compresses) is not the outer surface. Use the mean diameter (outer diameter minus material thickness) for more accuracy.
• Bending allowance: When forming, the material may stretch or compress slightly; a test piece is often recommended.
• Seam and joint allowances: These are not included. Add them to the straight edges.
• Precision: For large cones, small angular errors become large gaps. Use a sharp scribe and check the angle with a protractor.

For most hobby and light fabrication work, the formula is accurate enough. For critical aerospace or pressure-vessel work, consult engineering standards.

Frequently Asked Questions

Can I use this for a full cone?

Yes, but the formula divides by \(d_2 – d_1\), so if the top diameter is zero, you get a division by zero. Enter a very small top diameter (e.g., 0.1 mm) to approximate a full cone. For exact results, use the full cone formula: radius = slant height = \(\sqrt{(d_2/2)^2 + h^2}\), and angle = \((d_2/2) / R \times 180^\circ\).

How do I add a seam allowance?

Add the desired overlap width to one of the straight edges of the sector. For example, if you want a 10 mm overlap, draw the second straight edge 10 mm beyond the calculated angle line, or add 10 mm to the length of one straight edge. The calculator does not include this.

What if my cone has a slanted top or bottom?

This calculator assumes the top and bottom are parallel circles. If they are not parallel, the pattern is more complex and requires a different method, often using triangulation or CAD software.

Why is the sector angle not 360°?

Because the pattern is only a portion of a full ring. When you roll it, the two straight edges meet, and the angle between them is the sector angle. A full ring would make a flat disk, not a cone.

Can I use inches instead of mm?

Yes, as long as all three inputs are in the same unit. The calculator does not convert units; you must enter consistent values. For example, if you measure in inches, enter all three in inches.

What is the difference between slant height and height?

Height is the perpendicular distance between the two circular faces. Slant height is the distance along the cone’s surface from the bottom edge to the top edge. The slant height is always longer than the height (unless the cone is flat). The formula uses height to compute slant height.

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in d1 | Top diameter | mm | 100 | 0
in d2 | Bottom diameter | mm | 200 | 0
in h | Height | mm | 150 | 0
out R1 | Large radius (arc) | mm | (d2/2) * sqrt(1 + (2*h/(d2-d1))^2) | 2
out R2 | Small radius (arc) | mm | (d1/2) * sqrt(1 + (2*h/(d2-d1))^2) | 2
out theta | Sector angle | degrees | (d2/(2*R1))*180/pi | 2