How to construct development of a cone

How to construct development of a cone

At development of surfaces all its flat elements are combined with one plane. If development of a polyhedron is based, as its flat element serves each side. And at expansion of a curve surface for simplification of construction the polyhedron fits into it. Mathematically such development will be approximate, but at its performance according to drawings in engineering practice, it is rather exact.

It is required to you

  • Pencil, triangle, ruler, protractor, curves, compasses


1. At creation of development it is necessary to follow the basic rules: - the sizes of all elements have to have full size. - the area of development is equal to the area of the developed surface.

2. Example. Construct development of an inclined cone (figure 1). Enter a pyramid in the set conic surface. For this purpose divide a circle of the basis of a cone into arches 1 ₁ 2 ₁; 2 ₁ 3 ₁, etc. Having connected these points chords, receive the parties of the basis of a pyramid, and her side edges will be rectilinear forming, carried out through these points and top of S (S ₁).

3. Determine the full size of side edges of S2, S3, etc. by way of a rectangular triangle. For this purpose designate height of a frontal projection of a cone of h, at right angle to h postpone horizontal projections of edges of S ₁, 2 ₁, S ₁, 3 ₁, S ₁, 4 ₁. The received hypotenuses are also required full sizes (N of century) of edges of S2, S3, S4.

4. Edges of S1 and S5 are frontal straight lines, i.e. they are parallel to the frontal plane of projections P ₂, so on it they were designed full-scale: S ₂ 1 ₂ = N in, S ₂ 5 ₂ = N of century. The basis of a cone is located in the horizontal plane of projections P ₁ therefore chords were designed without distortion, i.e. these are their full sizes (N of century) – 1 ₁ 2 ₁; 2 ₁ 3 ₁, etc.

5. Development of a pyramid represents its sides combined with the plane of the drawing in the form of triangles. For their construction on any vertical straight line from S point ₀ postpone S₂1 piece ₂, S1 equal to full size of an edge. Of a point 1 ₀ make notches with a radius of 1 ₁ 2 ₁, and of S ₀ – S radius ₀ 2 ₀. 2 ₀ connect the received point straight lines to S ₀ and 1 ₀.

6. S triangle ₀ 1 ₀ 2 ₀ – one of sides of the entered pyramid. In this way construct adjacent sides and find points 3 ₀, 4 ₀, 5 ₀. Having connected them to S ₀, receive development of a side surface of a pyramid.

7. Then connect 1 ₀ 2 ₀ 3 ₀, 4 ₀, 5 ₀ a lekalny curve is and there will be a required development of the set conic surface. Development is symmetric rather direct S ₀ 1 ₀ since the surface has the symmetry plane.

Author: «MirrorInfo» Dream Team