Cone Pattern Calculator: How to Build a Flat Cone

A cone pattern calculator works out the flat layout dimensions you need to cut and roll into a cone: the arc radius, the arc angle, and the chord length. You enter the cone’s base diameter and its slant height (or the height and base diameter, from which it derives the slant height), and you get the full flat pattern geometry in seconds. This page explains the formulas, walks through a worked example, and covers the common mistakes that ruin otherwise good cone layouts.

Formula

The flat pattern of a right circular cone is a sector of a circle: a wedge with a curved outer edge and two straight sides that meet at the apex. The two straight sides are both equal to the cone’s slant height, and the curved edge length equals the circumference of the cone’s base. The three key quantities are:

• Slant height (s): the distance from the apex to the rim along the cone’s surface. For a right cone, s = √(r² + h²), where r is the base radius and h is the perpendicular height.
• Arc radius: the radius of the flat sector, which is exactly the slant height s.
• Arc angle (θ): the angle of the wedge, in degrees. Because the arc length of the sector equals the base circumference, θ = (base circumference / arc radius) × (180/π) = (2πr / s) × (180/π) = 360 × (r / s).
• Chord length: the straight-line distance between the two ends of the arc, useful for marking the layout on a flat sheet. Chord = 2 × s × sin(θ/2).

All lengths are in the same unit (mm, cm, inches, etc.), and the angle is in degrees. The calculator above takes base diameter and height, computes the slant height, then derives the arc angle and chord length.

Worked Example

Using the default values from the calculator: base diameter d = 100 mm, height h = 150 mm.

1. Base radius r = d/2 = 50 mm.
2. Slant height s = √(50² + 150²) = √(2500 + 22500) = √25000 = 158.1139 mm.
3. Arc angle θ = 360 × (r / s) = 360 × (50 / 158.1139) = 360 × 0.31623 = 113.84°.
4. Chord length = 2 × s × sin(θ/2) = 2 × 158.1139 × sin(56.92°) = 316.2278 × 0.8381 = 265.0 mm.

The calculator rounds the slant height to 158.1 mm, the arc angle to 113.8°, and the chord to 265.0 mm. When you draw this sector on a flat sheet, the two straight edges are 158.1 mm long, and the curved edge, when rolled, forms a base with a 100 mm diameter.

When to Use a Cone Pattern Calculator

You need a cone pattern calculator any time you are fabricating a cone from a flat material: sheet metal, cardboard, plastic, fabric, or paper. Common real-world uses include:

• Making a funnel, a speaker horn, or a decorative cone.
• Fabricating a transition piece for ductwork or a hopper.
• Cutting a pattern for a party hat, a cone for crafts, or a model rocket nose cone.
• Working out the layout for a truncated cone (also called a frustum) – though that requires a slightly different calculation, often done with a dedicated frustum calculator.

The calculator is for a right circular cone, meaning the apex is directly above the center of the base. If your cone is oblique (apex off-center), the pattern is more complex and requires a different method.

How to Read the Results

The output gives you three numbers to transfer to your material:

• Arc radius (slant height): set your compass or trammel to this length. This is the distance from the apex point to the curved edge.
• Arc angle: the total angle between the two straight edges. You can mark this with a protractor, or you can use the chord length to set the gap between the two ends of the arc.
• Chord length: the straight-line distance between the two ends of the arc. If you draw the arc with a compass, measure the chord along the arc’s edge to find where the second straight edge meets it.

When you cut out the sector, the two straight edges meet at the apex. Overlap them to form the cone, and the curved edge becomes the base circle. The seam can be taped, welded, glued, or riveted depending on the material.

Common Mistakes and How to Avoid Them

Several errors routinely produce misshapen cones:

• Using the height instead of the slant height: the arc radius is the slant height, not the vertical height. If you use the vertical height, your cone will be too steep and the base will not close.
• Mixing units: if you enter the diameter in inches and the height in millimeters, the slant height will be wrong. Keep everything in the same unit.
• Forgetting to halve the diameter: the formula uses the radius, so if you enter the diameter directly into the arc angle formula, you get double the correct angle.
• Measuring the chord along the curve instead of straight: the chord is a straight line between two points on the arc, not the length of the arc itself. If you measure the curved distance, you will get the wrong gap.
• Rounding too early: if you round the slant height to 158 mm and then compute the angle, the error is small, but if you round the angle to 114° and then compute the chord, the chord can be off by a few millimeters on a large cone. Keep full precision until the final step.

Limits of the Formula

The formula assumes a perfect right circular cone with a sharp apex and a flat base. Real fabrication introduces material thickness: if you are working with thick sheet metal, the neutral axis (the line that neither stretches nor compresses) is not at the surface, so the pattern may need adjustment. For thin materials like paper or light cardboard, the error is negligible. Also, the formula gives the pattern for the outside surface; if you need a cone with a flange or a hem, you must add that allowance separately. For a truncated cone (frustum), the flat pattern is an annular sector – a sector with a smaller arc cut out of the middle – and the calculation is different.

Frequently Asked Questions

What is the difference between slant height and height?
Height is the perpendicular distance from the base to the apex, measured straight up. Slant height is the distance along the cone’s surface from the apex to the base edge. For a right cone, slant height is always longer than height, and it is the radius of the flat pattern.

Can I use this for a cone with a flat bottom (truncated cone)?
No. A truncated cone has a smaller circle at the top, and its flat pattern is an annular sector. You need a frustum calculator, which takes the top diameter, bottom diameter, and height, and returns the two arc radii and the angle.

What if my cone is not right (apex not centered)?
An oblique cone has an apex that is not over the base center. Its flat pattern is not a simple sector; it requires triangulation or numerical methods. This calculator only handles right circular cones.

How do I mark the arc on a large sheet without a giant compass?
Use a trammel: a straight stick with a pencil at one end and a nail or pin at the distance of the arc radius. Or use a string tied to a pencil, with the pin at the apex point. For very large radii, you can calculate a series of points along the arc using the angle and radius, then connect them with a smooth curve.

What is the formula for the arc length?
Arc length = (θ/360) × 2π × s, where θ is the arc angle in degrees and s is the slant height. This equals the base circumference, 2πr, which is the check that your pattern is correct.

Can I use this for a cone with a thick material?
For thick materials, the pattern may need adjustment for the material’s thickness, especially if you are bending metal. The neutral axis method is used in sheet metal work; for most hobby projects with thin material, the simple formula is accurate enough.

Related

• Multiple Regression Calculator 3 Variables
• Linear Regression Power Calculator: How to Use It
• Linear Regression Test Calculator: How It Works

in d | Base diameter | mm | 100 | 0
in h | Cone height | mm | 150 | 0
out s | Slant height | mm | sqrt((d/2)^2 + h^2) | 1
out r | Arc radius (slant height) | mm | s | 1
out a | Arc angle | degrees | (d / (2 * s)) * 360 | 1
out chord | Chord length | mm | 2 * s * sind(a / 2) | 1