Development of Cones

DEVELOPMENT OF CONES

Development of lateral surface of a cone is drawn by radial line method. The lateral development of cone is in the form of sector of a circle, the radius of which is equal to the slant height of the cone.

Consider a cone of radius of the base circle `r' and the altitude 'h' as shown in Fig. 16.28(i). Let 'l' be the slant height of the cone. The development of lateral surface of the cone is shown in Fig. 16.28(ii), it is sector of circle with radius 'l' (ie, the slant height of the cone) and subtended angle `θ' at centre.

Hence to draw the development of lateral surface of a cone, locate the centre point O' first, and draw an arc of radius 'l' (ie. slant height) with O' as centre. Then draw two radii preferably keeping one of the radii as straight at an included angle of θ°. If the circumference of base circle is divided into 12 equal parts, the arc drawn in the development may also be divided into 12 equal parts.

Note: Alternatively the length of arc in development may be transferred from the base circle of cylinder by measuring 1/12th part and marking 12 times on the arc in the development.

The development of cone with cut-out/ hole may be drawn in similar way as explained for prisms and cylinders.

Example 23: Draw the development of the lateral surface of a cone of base diameter 48 mm and altitude 55 mm.

1. Draw the top view as a circle of diameter 48 mm and divide it into 4 equal parts. (Here no cutting plane is given which cut the generators, hence the circumference of circle is divided into minimum number of divisions, whereas for the problems like truncated cone (or) cone with cutout / hole it is suggested to divide the circumference of base circle into 8 (or) 12 equal parts to get sharp curve)

2. Name the points as a, b, c and d.

3. Project the top view and complete the front view.

4. The end generators o'a' and o'c' give the true length of the generators because their top views are parallel to XY. Measure the slant length o'a'.

5. Determine the angle subtented at the center of sector of a circle, which is the development of lateral surface of cone.

6. Locate the point O and draw OA of length equal to the slant length of cone (equal to o'a' in front view) and draw another line OA at an angle of 144°. 

7. With O as centre and OA as radius draw an arc. Divide the arc A-A into 4 equal parts and name the parts as A, B, C, D, A which is the required development.

Note: If truncated cone (or) cone with cutout / hole are given, draw the development of full cone first and then transfer the cutting plane / cut out / hole to the development.

Example 24: Draw the development of the lateral surface of frustrum of cone of base diameter 60 mm, top base diameter 25 mm and altitude 50 mm.

1. Draw the top view and front view of the frustum of cone and name the points accordingly.

2. Extend a'e' and c'g' to intersect at o', which is the apex of the cone, and measure the slant length l, equal to o'c'.

3. Determine the angle θ, θ

4. Locate the point O and draw two lines OA equal to the true slant length o'a' at subtended angle of θ = 114°

5. Mark OE on the line OA such that OE = o'e'.

6. With O as centre draw two arcs in between the lines OA with radius o'e' and o'a'.

7. Divide the arc AA into 4 equal parts and join the points with O.

8. Darken the edges E-A-A-E-E. which is the required development.

Example 25: A cone of 50 mm base diameter and 70 mm long axis, rests on its base on HP. Draw the development of its lateral surface when it is cut by an auxiliary inclined plane bisecting the axis and inclined at 45° to HP.

1. Draw a circle of diameter 50 mm as top view and divide the circumference of circle into 12 equal parts.

2. Project the top view and complete the front view as a triangle of height 70mm.

3. Draw the generators in front view for the points b, c, d etc. using thin lines.

4. Draw V.T of cutting plane, inclined at 45° to XY and bisecting the axis.

5. Mark the points of intersection of cutting plane with the generators as 1', 2', 3' etc.

6. The top view of end generators o'a' and o'g' are parallel to XY, hence o'a' (or o'g') gives inale the true slant length. 

7. The top view of other generators are not parallel to XY hence through the points 2', 3' etc. draw horizontal lines to cut the true slant length o'a' at 21', 31' etc. Now the lengths 0'11', 0'21' etc. represents the true length. 

8. Mark the point O and draw a line OA equal to the true slant length 'o'a'.

9. Determine the angle

10. Draw another line OA at angle of 121° from the line OA drawn earlier.

11. With O as centre and OA as radius draw an arc, touching both OA lines.

12. Divide the arc AA into 12 equal parts and Join the points with O, which represents the generators of cone.

13. Transfer the distances 0'21', 0'31', etc. on the respective generators in the development to get the points 1, 2, 3 etc. respectively.

14. Join the points 1, 2, 3 etc. using thick line, which is the required development.

Example 26: A cone of 50 mm base diameter and 70 mm long axis rests on its base on HP. Draw the development of its lateral surface when it is cut by an auxiliary inclined plane bisecting the axis and inclined at 60° to HP.

The development is shown in Fig. 16.32.

Example 27: A cone is cut by two cutting planes one is parallel to HP bisecting the axis and the other one is inclined at 30° to HP, passing through one extremity of the base. Draw the development of the lateral surface of cone in between these two cutting planes. Take the diameter of base and axis length as 60 mm and 80 mm respectively.

The development is shown in Fig. 16.33.

Example 28: A cone of base diameter 50 mm and altitude 70 mm is lying on its base on HP. It is sectioned by a plane normal to both HP and VP at a distance of 10 mm to right of the axis of the cone. Draw the development of the lateral surface of the left side remaining portion.

The development is shown in Fig. 16.34.

Example 29: A cone of base diameter 60 mm and axis 75 mm long is resting on its base on HP. A profile section plane (normal to both HP & VP) cuts the cone at 20 mm from the left end of cone. Draw the development of the retained cone.

The development is shown in Fig. 16.35.

Example 30: A right circular cone of diameter 50mm axis 60mm is resting on the ground on its base. Draw the development of lateral surface. Determine the shortest length of a string required to wound round the lateral surface of the solid starting from one extreme point and ending at the same point. Also trace the points on to the projections.

1. Draw the top view, front view and development of cone.

2. Divide the circumference of circle in top view into eight equal parts and name the points as a, b, c etc., and transfer these points to front view and development.

3. Join the end points on the development A-A with a straight line which gives the shortest length of string. It is measured from development as 170 mm.

4. Let the line joining the points A-A in development cut the generators at 1, 2, 3 etc., Transfer the lengths 0-1, 0-2 etc. to front view (for example mark the length of 0-1 from development on the extreme generator o'a' in front view as 1'1 and draw a horizontal projector through 1'1 to cut o'b' generator at 1').

5. Join the points 1', 2' etc. by a smooth freehand curve which gives the front view with the string. 

6. Project the points 1', 2' etc. to the top view (for example draw a vertical projector through 1' to cut ob line in top view at 1).

7. Join these points 1, 2, 3 etc. by a smooth free hand curve which gives the top view with the string.

Example 31: A cone of 60 mm base diameter and 75 mm long axis is resting on its base on HP. A square hole of 25 mm side is made in it such that the axis of cut out is perpendicular to VP, intersects the axis of cone at a height of 25 mm from the base and the faces of the hole are equally inclined to HP. Draw the development of its lateral surface.

The lateral development is shown in fig. 16.37.

Example 32: A cone of base diameter 80 mm and axis 80 mm long is resting upon HP on its base. A horizontal square hole of 40 mm side is cut in the cone in such a way that the axis of the hole intersects the axis of the cone at a height of 16 mm from the base and four side of the hole are equally inclined to HP. Draw the development of the lateral surface of the cone.

The development is shown in Fig. 16.38.

2. Development of cone with cylindrical hole

Example 33: A cone of 50 mm base diameter and axis 70 mm long stands on its base on HP. A circular hole of 15 mm radius is drilled through the axis of the cone such that the axis of hole is perpendicular to the axis of cone and the centre of hole is at 20 mm above base. Draw the development of the lateral surface of the cone with hole.

The development is shown above in Fig. 16.39.

Note that the ends of horizontal diameter of hole in front view touches the generators o'b' and o'h'.

Hint: Draw the circle of hole of 15 mm radius in front view keeping its centre point 20 mm above base. Mark the points of intersection of circumference of circle with generators. Transfer these poins in the development on the corresponding generators.