Cone Development Calculator
About Cone Development Calculation
This calculator computes the geometry of the development of a right cone and a frustum from the given linear dimensions in millimetres. The result is used for making flat patterns from sheet materials, marking out parts, preparing drawings, and checking the dimensions of the lateral surface before cutting or forming.
The calculation is based on the geometric development of the surface and helps obtain a sector of a circle or an annular sector which, after rolling, forms a cone of the required shape. This approach is suitable for working with metal, plastic, cardboard, and other materials when a flat pattern is required without accounting for material thickness.
Reference points and recommendations
Right cone
Input data. For a right cone, the calculation uses the base diameter d in mm and the height h in mm. The calculator first determines the base radius r = d / 2, then calculates the slant height, which is the distance from the cone apex to the edge of the base along the lateral surface.
L = √(h2 + r2)
Meaning of L. The value L in mm is the radius of the sector that must be drawn on a flat sheet to obtain the development of the lateral surface of the right cone. This radius is used to draw the arc of the pattern.
Development angle. After finding L, the calculator determines the central angle of the sector so that the arc length of the sector is equal to the circumference of the cone base. For this, it uses the ratio between the base circumference πd and the full circumference with radius L, which is 2πL.
α = 360° × d / (2L)
Meaning of α. The value α in degrees shows which sector of the circle must be cut out. If a sector with radius L and angle α is drawn, its arc will match the circumference of the base after rolling.
Frustum
Input data. For a frustum, the calculation uses the height h in mm, the lower diameter d1 in mm, and the upper diameter d2 in mm. The calculator first treats the frustum as part of a full cone and extends it to the apex.
Extension to a full cone. The base radii are taken as r1 = d1 / 2 and r2 = d2 / 2. Then, using triangle similarity, it determines the full height of the imaginary cone H, from the apex to the larger base.
H = h × r1 / (r1 - r2)
Outer and inner development radii. After that, the calculator computes the outer slant height R2 and the inner slant height R1. These values in mm become the two radii of the annular sector used for the development.
R2 = √(H2 + r12)
R1 = √((H - h)2 + r22)
Meaning of R1 and R2. The value R2 defines the outer arc of the pattern, and R1 defines the inner arc. The radial distance between them is equal to the slant height of the frustum and forms the width of the annular sector.
L = R2 - R1
Development angle. The central angle for the frustum is selected so that the outer arc of the development is equal to the circumference of the larger base. This gives one common angle for both arcs of the annular sector.
α = 360° × d1 / (2R2)
Consistency check. The same angle automatically gives an inner arc corresponding to the circumference of the upper base. This happens because of geometric similarity, so the calculator does not choose between several angles but obtains one single value that fits both edges at the same time.
Assumptions used
Geometric model. The calculation is carried out for an ideal circular cone or an ideal circular frustum. It is assumed that the axis passes through the centre of the base and that both bases are perpendicular to the axis.
Developable surface. The calculator works only with the lateral surface. Material thickness, seam allowance, overlap, cut width, edge forming, elastic deformation, and manufacturing losses are not included in the result and must be added separately according to the production method.
Units and rounding. All linear dimensions are calculated in millimetres, and the angle is calculated in degrees. If only whole numbers are displayed on the page, this is only a display format. In practical layout work, an allowance of 1-3 mm is often used for marking and final fitting when the part is made manually.
Standards and drawing references
Dimensioning. For dimensioning and drawing presentation, a common reference is EN ISO 129-1 Technical product documentation. Presentation of dimensions and tolerances. Part 1. General principles. This standard is useful for the correct notation of diameters, radii, angles, and linear dimensions on the drawing of the development.
Lines and graphics. For line types, arcs, leaders, and general graphical presentation, a common reference is EN ISO 128-2 Technical product documentation. General principles of representation. Part 2. Basic conventions for lines. It helps show the part outline, centre lines, and dimension elements correctly.
Projection methods. When the development is included in a technical drawing set, the EN ISO 5456 Technical drawings. Projection methods series is used as a general reference. These documents relate to the rules for representing shape on drawings rather than to the development formula itself, so the calculator performs a geometric calculation, not a standards-based design check.
FAQs
Why does a right cone need only one development radius?
Because the lateral development of a right cone is a sector of a circle. All points of the base edge are at the same distance from the apex, and this distance is equal to the slant height L.
Why does a frustum development use two radii?
The development of a frustum has the shape of an annular sector rather than a simple sector. That is why it needs the inner radius R1 and the outer radius R2, with the lateral surface located between them.
Is this cone development calculation suitable for sheet metal with thickness?
Yes, as a basic geometric calculation of a cone development it is suitable for sheet metal. For real cutting layouts, however, thickness, joining allowance, seam type, and forming method are usually added separately because they change the working contour of the blank.
Can this calculation be used to make a pattern from paper, cardboard, or plastic?
Yes, this cone development calculator is suitable for any material that can be cut as a flat template and then rolled into shape. For flexible materials, the result is usually used directly, while for rigid materials additional manufacturing corrections are added.
Why is the development angle less than 360°?
Because the lateral surface of a cone occupies only part of a full circle with the slant height as radius. The angle is chosen so that the arc length of the sector exactly matches the circumference of the base, otherwise the cone will not close to the correct size after assembly.