How to use the cone flat pattern calculator
- Enter the top diameter and the bottom diameter of the cone. Use 0 for the top if the cone comes to a point.
- Enter the straight height, measured straight up from the bottom opening to the top, not along the sloping side.
- Choose the unit that matches your measurements. It is only a label for the results.
- Draw two arcs from one point with the outer and inner radii, cut out a piece with the sector angle between two straight lines, and roll it into a cone.
Formula
slant height s = sqrt(height^2 + ((bottom - top) / 2)^2)
outer radius R = bottom x s / (bottom - top), inner radius r = top x s / (bottom - top)
sector angle = 180 x (bottom - top) / s degrees
surface area = pi x (top + bottom) / 2 x s
Unrolling the sloping side of a cone gives part of a ring. Its outer arc is as long as the bottom circle (pi times the bottom diameter) and its inner arc is as long as the top circle.
A funnel shape 4 inches wide at the top
Take a top diameter of 4, a bottom diameter of 10 and a height of 8, all in inches. The slant height is sqrt(8^2 + 3^2) = 8.544. The outer radius is 10 x 8.544 / 6 = 14.24 and the inner radius is 4 x 8.544 / 6 = 5.696, so the two arcs are drawn from a point 14.24 and 5.696 inches away.
The sector angle is 180 x 6 / 8.544 = 126.40 degrees. The outer arc is pi x 10 = 31.42 inches long and the inner arc is pi x 4 = 12.57 inches. The flat piece fits in a box 25.42 by 11.67 inches, and its area is 187.89 square inches, the same as the area from the sector formula.
A pointed cone 12 inches across and 9 inches tall has a slant height of 10.82 inches, no inner arc, and a sector angle of 199.69 degrees.
Making the piece
Draw a long straight line, mark the apex, and swing the outer and inner arcs with a string compass or a trammel. Lay out the sector angle with a protractor or by measuring the outer chord, which is 2 x R x sin(angle / 2). Add a seam or overlap allowance outside the lines and make a paper test piece first.
To find how much the finished cone holds, use the Volume Calculator. For arcs and radii of a curve through points, use the Circle Calculator.
When the diameters are equal
If the top and bottom diameters match, the shape is a straight tube. It unrolls into a rectangle as wide as the circumference and as tall as the height, so there is no sector to draw.
More shape and maker tools
For other shapes, patterns and maker projects, try the Circle Skirt Calculator, Circle From Three Points Calculator and Polygon Area Calculator.
Assumptions and limits
- The cone is a right circular cone with straight sloping sides.
- The material is thin, flat and does not stretch.
- Seams, overlaps and rolled edges are not included.
- The height is the straight height between the two circles.
Frequently asked questions
How do I make a cone template from a flat sheet?
Find the slant height from the straight height and the diameters, draw arcs for the outer and inner radius from one point, and cut a sector with the angle the tool gives. Roll it so the straight edges meet.
What is the sector angle of a cone pattern?
It is 180 times the difference of the diameters divided by the slant height. A 4 to 10 inch frustum 8 inches tall has a sector of 126.40 degrees.
Why can the angle be more than 180 degrees?
A tall, wide cone is flat enough that the unrolled piece is more than half a circle. The tool still gives the right box size for cutting the piece out.
Does the pattern include an overlap for glue or rivets?
No. The lines are the finished edges, so add your seam allowance on the outside of the pattern.