Cone and Frustum Development

Developing a cone

A right cone develops into a circular sector. A concentric frustum develops into an annular sector. First obtain slant height from axial height and radius difference. Similar triangles give the pattern radii, and matching arc length to the original circumference gives the sector angle.

Example and scope

A cone of radius 300 mm and height 400 mm has slant height 500 mm and sector angle 216 degrees. Its inner pattern radius is zero. Dimensions describe the neutral surface. Use cylinder development for equal radii. No drawing or cutting file is generated and seam allowance must be prepared separately.

Check the annular sector by hand

Use a large neutral radius of 500 mm, small neutral radius of 200 mm and axial height of 400 mm. The radius difference is 300 mm and slant length is √(400² + 300²) = 500 mm. The outer development radius is 500 × 500 / 300 ≈ 833.333 mm and the inner radius is 500 × 200 / 300 ≈ 333.333 mm. The sector angle is 360 × 300 / 500 = 216°. Net lateral area is π × (500 + 200) × 500 / 1000000 ≈ 1.099557 m².

The difference between development radii must equal the slant length. The outer arc length equals the large base circumference, while the inner arc equals the small base circumference. These identities do not detect incorrectly entered radii. Independently measure the large base circumference on the source drawing and compare it with the outer arc length. Do not enter a length measured along the sloping side as the axial height.

The annular sector area differs from its enclosing rectangle, so purchasing quantity depends on nesting. As the base radii approach each other, the development radii become large even without a tall cone. In this nearly cylindrical case, examine measurement precision and the intended manufacturing method separately.

Frequently asked questions

How do you calculate a frustum flat pattern?

For an example frustum with neutral surface radii 300 and 100 mm and axial height 400 mm, the flat pattern has outer radius 670.8 mm and sector angle 161°. The annular sector follows from similar triangles and equal circumferences, with seam allowances and cutting waste excluded.

Calculate with these values