How to Build a Straight Truncated Paper Cone from Horizontal Sections

A horizontal section is a slice parallel to the bottom rim. For a right circular conical frustum, its radius changes linearly with height. Consequently, sections at equal height intervals must have equal changes in diameter. This follows from the radius-versus-height relationship given in Wolfram MathWorld’s “Conical Frustum”.

Start by recording each section’s height above the bottom and its diameter. Keep the units consistent, and measure height vertically along the axis rather than along the sloping side.

Here is the illustrative section schedule used throughout this guide:

Height above bottom	Section diameter	Section radius
0 mm	120 mm	60 mm
10 mm	105 mm	52.5 mm
20 mm	90 mm	45 mm
30 mm	75 mm	37.5 mm
40 mm	60 mm	30 mm

Each 10 mm rise reduces the diameter by 15 mm, so these sections describe a straight taper. If your sections have unequal spacing, compare diameter change per millimeter of height instead. Those rates must agree for a single straight cone.

The section centers must also lie on one vertical axis. If the centers shift sideways, or the diameter changes at different rates, the simple concentric-arc pattern below will not reproduce the supplied shape. Resolve those differences before drawing your final pattern.

For the initial model, gather paper that rolls easily, a pencil, ruler, compass, protractor, scissors, and paper adhesive or double-sided tape. Use a sheet large enough for the complete pattern and its joining tab. A scrap-paper trial helps you check the layout before using your chosen finish.

Define these dimensions:

• R: radius of the larger, bottom opening.
• r: radius of the smaller, top opening.
• h: vertical distance between those openings.

For the example, R = 60 mm, r = 30 mm, and h = 40 mm. These are ideal surface dimensions: the calculation treats paper thickness as negligible. If the finished shell must fit closely around another object, test it with the actual paper and adjust the fit before making the final version.

Draw a full-size side elevation: a centered trapezoid with a 120 mm lower edge, a 60 mm upper edge, and 40 mm between them. Extend both sloping sides upward until they meet. That intersection is the missing cone’s apex. Drawing the elevation with its apex is also the starting point in Brian Coey’s “Frustum of a Cone” construction lesson.

The distance down the sloping side is longer than the vertical height. Call this slant distance s. For a right circular frustum, s = √[h² + (R − r)²], as documented in Wolfram MathWorld’s frustum geometry.

For the example:

s = √[40² + (60 − 30)²] = √2500 = 50 mm.

The pattern needs two radii measured from a shared center, O. These are distances from the original apex along the cone’s surface—not the radii of its circular openings. Call the larger pattern radius L and the smaller one l.

By similar triangles, L/l = R/r. Their difference is the frustum’s slant distance, so L − l = s. Solving those two relationships gives:

• L = R × s ÷ (R − r)
• l = r × s ÷ (R − r)

Therefore, the example needs L = 60 × 50 ÷ 30 = 100 mm, and l = 30 × 50 ÷ 30 = 50 mm. Their 50 mm difference matches the calculated slant distance.

Next calculate the retained sector angle, θ:

θ = 360° × R ÷ L = 360° × (R − r) ÷ s.

This angle makes the outer pattern arc equal to the bottom circumference. NPTEL’s original “Development of a Right Circular Cone” lesson gives the same circumference-based sector construction. Removing the small cone’s sector leaves the frustum pattern.

For this example, θ = 360° × 60 ÷ 100 = 216°. Keep the 216° portion of the ring. The missing wedge is 360° − 216° = 144°. Label both values on your drawing so you do not accidentally keep the smaller wedge.

Your intermediate sections give you useful registration lines. At height z above the bottom, the section radius is:

a(z) = R − (R − r) × z ÷ h.

Using the same similarity ratio as the rim calculations, its corresponding distance from pattern center O is:

ρ(z) = a(z) × s ÷ (R − r).

The following values are calculated from the illustrative dimensions. They specify where each horizontal section appears on the flat sheet:

Height above bottom	Finished section diameter	Pattern arc radius from O
0 mm	120 mm	100 mm
10 mm	105 mm	87.5 mm
20 mm	90 mm	75 mm
30 mm	75 mm	62.5 mm
40 mm	60 mm	50 mm

Draw the intermediate arcs lightly at 87.5, 75, and 62.5 mm from O. After assembly, they represent horizontal levels 10, 20, and 30 mm above the bottom. Keep these as pencil guides; cutting them would divide the shell into separate bands.

Notice that a 10 mm vertical interval becomes a 12.5 mm interval along the flat pattern’s radial direction. That difference is why transferring vertical section spacing directly onto the sloping surface produces the wrong height.

Use the calculated angle to draw the exact sector rather than estimating its length with repeated compass steps. Coey’s “Radial Line Pattern Development” explains the circumference requirement and notes that step-off methods introduce accuracy limitations.

1. Mark center O with enough space around it for a 100 mm radius.
2. Draw concentric circles or arcs of radius 100 mm and 50 mm.
3. Draw two radial boundary lines enclosing the retained 216° sector. With a 180° protractor, lay out the complementary 144° wedge and label it for removal.
4. Add the three intermediate section arcs from the table.
5. Keep the ring-shaped area between the 50 mm and 100 mm arcs within the retained sector.
6. Add a joining tab outside one radial boundary. For this trial model, an approximately 8 mm tab is a practical starting choice, not part of the geometric calculation.
7. Cut the outer arc, inner arc, opposite radial edge, and tab outline.

Keep the original radial boundary visible where the tab joins the pattern. This is the seam registration line. Trim the tab ends back slightly from both rim arcs so the tab does not protrude past the finished openings.

The two concentric arcs are essential: Coey’s frustum construction likewise uses the large and small slant lengths from a common radius point. A trapezoid copied directly from the side elevation is not the developed surface.

Gently curl the paper across the width of the sector, encouraging a gradual curve. Avoid creasing the main surface: the finished shell should curve around its circumference while remaining straight along each sloping line from rim to rim.

Bring the un-tabbed radial edge onto the registration line at the base of the tab. Place the tab beneath the opposite edge. Temporarily secure the seam, then check that the inner and outer arc endpoints meet and that each intermediate section line continues across the join.

Do not overlap the main pattern beyond its registration line to make the seam look tighter. The retained sector already contains the required circumference; additional overlap reduces the opening sizes. Apply adhesive to the tab only, align the seam again, and hold it according to the adhesive’s instructions.

For this worked example, the calculated outer arc length is approximately 376.99 mm, matching π × 120 mm. The inner arc is approximately 188.50 mm, matching π × 60 mm. These calculated checks confirm the pattern dimensions; they are not measurements of a finished model.

Set the larger rim on a flat surface and check the perpendicular height to the smaller rim. For the example, that height should be about 40 mm, while the distance along the sloping surface should be about 50 mm. Measure each opening through its center in more than one direction to see whether it is circular.

What you observe	Check next
Both openings are too small	Confirm the retained angle and check for excess seam overlap.
Both openings are too large	Check for a seam gap or an oversized retained angle.
Sloping distance is correct but vertical height is wrong	Recheck both opening diameters and whether the rims are parallel.
Section lines do not meet at the seam	Check that the radial edges align without sliding lengthwise.
The shell has corners or flat panels	Check whether the paper was creased along construction lines.

If R equals r, the shape is a cylinder and these formulas divide by zero; use a rectangle whose length is the circumference and whose width is the height. If r equals zero, the smaller opening disappears and the pattern becomes a full cone sector. With a very slight taper, R − r becomes small and the pattern center lies far away, making manual compass layout cumbersome. The BCcampus radial-line lesson identifies a reasonably accessible radius point as a practical condition for this construction method.

A successful build starts with consistent horizontal sections, uses slant distances for the flat pattern, and closes at the marked seam. Keep the section schedule and pattern dimensions together so you can reproduce the cone or change its size without reconstructing the geometry.

• Frustum of a Cone — Pattern Development: Sheet Metal Level 1, Brian Coey
• Radial Line Pattern Development — Pattern Development: Sheet Metal Level 1, Brian Coey
• Lecture 36: Development of Surfaces of Truncated Solids — Development of a Right Circular Cone, NPTEL
• Conical Frustum — Wolfram MathWorld