Cone Development Calculator
About Cone Development Calculation
The cone development calculator determines the geometric dimensions of a flat pattern for a right circular cone or a truncated cone (frustum). It calculates the development radii, central angles, slant height, chord of the outer arc, and lateral surface area.
The dimensions can be used to make components from sheet metal, plastic, cardboard, and other materials. They allow you to draw a circular sector or annular sector that forms a cone of the specified dimensions when rolled up.
Reference information and recommendations
Geometric basis of the calculation
Development principle. When the lateral surface of a right circular cone is unfolded onto a plane, it forms a circular sector. A truncated cone produces an annular sector bounded by two concentric arcs. The length of each arc must equal the circumference of its corresponding base.
Input dimensions. The calculation uses the cone height and the diameters of its bases. All linear dimensions are entered in millimetres. For a truncated cone, the larger diameter is d1 and the smaller diameter is d2. The condition d1 > d2 must be satisfied.
Development of a right circular cone
Slant height L. This is the distance from the cone apex to the edge of the base along the lateral surface. It is calculated using the Pythagorean theorem from the height h and the base radius d / 2:
L = √(h2 + (d / 2)2)
The resulting value L, in millimetres, is also the radius of the circular sector forming the development.
Development angle b. The central angle of the sector is found from the condition that its arc length equals the base circumference, π × d. Since a full circle of radius L has a circumference of 2 × π × L, the angle is calculated as follows:
b = 180° × d / L
Angle a. This is the angle of the remaining part of the full circle, which is not included in the development. It complements angle b to 360°:
a = 360° - b
To draw the flat pattern of a right circular cone, use radius L and sector angle b. Angle a indicates the size of the part excluded from the full circle.
Development of a truncated cone
Slant height L. First, the distance between the upper and lower circles along the inclined lateral surface is determined. This uses height h and half the difference between the base diameters:
L = √(h2 + ((d1 - d2) / 2)2)
Outer radius R2. The truncated cone is conceptually extended to the apex of the complete cone. Similar triangles are used to find the distance from that apex to the edge of the larger base:
R2 = L × d1 / (d1 - d2)
Inner radius R1. The distance from the apex of the complete cone to the edge of the smaller base is determined in the same way:
R1 = L × d2 / (d1 - d2)
The difference between the outer and inner radii equals the slant height of the truncated cone:
R2 - R1 = L
Development angle b. Both arcs of the annular sector share the same central angle. It is calculated from the difference between the diameters and the slant height:
b = 180° × (d1 - d2) / L
At this angle, the outer arc has length π × d1, while the inner arc has length π × d2. Thus, both edges of the development match the circumferences of the truncated cone's bases.
The angle of the excluded sector is calculated in the same way as for a right circular cone:
a = 360° - b
Chord of the outer arc
Chord C. This is the straight-line distance between the endpoints of the development's outer arc. It is used to check the geometry of the flat pattern and the positions of its endpoints during marking out.
The chord is calculated from the radius of the outer arc and half of central angle b:
C = 2 × R2 × sin(b / 2)
For a right circular cone, the slant height L is used instead of R2. Angle b is expressed in degrees in this formula.
The chord is always a straight line segment. Its length differs from that of the corresponding curved arc.
Development area
Area S. The area of the flat development equals the lateral surface area of the original cone. For a right circular cone, it is calculated from the base diameter and slant height:
S = π × d × L / 2
For a truncated cone, the sum of the two base diameters is used:
S = π × (d1 + d2) × L / 2
Both formulas initially give an area in mm2. To convert the result to m2, divide it by 1 000 000.
Assumptions and precision
Geometric model. The calculation assumes an ideal right circular cone. For a truncated cone, the bases are assumed to be parallel and coaxial, with their centres on the same straight line.
Adjustment factors. No additional allowance factors or manufacturing margins are applied. The calculation uses the specified geometric dimensions without adjustments for material thickness, cutting kerf, or joint seam.
Result precision. Linear dimensions in millimetres and angles in degrees are displayed to 0.1, without unnecessary trailing zeros. Areas of 0.01 m2 or more are rounded to two decimal places, and smaller areas to four. Intermediate calculations are performed without prior rounding.
European standards for technical drawing
EN ISO 129-1 - Technical product documentation. Presentation of dimensions and tolerances. Part 1: General principles. This standard sets out the rules for indicating dimensions and tolerances on technical drawings. It is relevant when showing radii, diameters, angles, and other dimensions of the development.
EN ISO 128-2 - Technical product documentation. General principles of representation. Part 2: Basic conventions for lines. This standard defines line types and their use in technical drawings, including dimension and extension lines.
These standards concern the presentation of technical documentation. The geometric dimensions of the development are calculated by mathematical formulas and do not depend on the drafting standard used.
FAQs
Why does the cone development calculation give two angles?
Angle b defines the sector that forms the cone's lateral surface. Angle a corresponds to the remaining part of the full circle. Together they equal 360°, so you can draw the development using angle b or mark angle a as the sector to be removed.
Why are the radii of a truncated cone development sometimes very large?
When the base diameters are close to each other, the lateral surface of the truncated cone approaches a cylindrical shape. In this case radii R1 and R2 increase while central angle b decreases. This is a normal geometric property of the development, not a calculation error.
Can I draw the cone development manually using the calculated dimensions?
Yes. For a right circular cone, draw a sector with radius L and angle b. For a truncated cone, draw two concentric arcs with radii R1 and R2, bounded by the same central angle b. Chord C provides an additional check of the straight-line distance between the outer arc's endpoints.
Why is the chord shorter than the arc of the development?
A chord joins two points on a circle with a straight line, whereas an arc follows a curved path between them. Therefore, a chord cannot be longer than the arc with the same endpoints. When making a cone, it is the arc length that must match the base circumference.
How can I determine the sheet material area needed to make a cone?
The calculated development area gives the cone's lateral surface area in m2. It equals the geometric area of the circular or annular sector. When selecting a sheet, also consider its overall dimensions and the pattern layout, because the area of a rectangular sheet may exceed that of the development itself.