# Problem 8.1 Development of Surfaces of Solids

## Problem 8.1 Development of Surfaces of Solids – A cone made up of Aluminium sheet with base circle diameter 65 mm and axis length 75 mm is kept on its base on the ground. A circular hole of 30 mm diameter is cut through the cone such that its axis remains perpendicular to V.P.; 10 mm to the right of the axis of cone and 25 mm above the base of cone. Develop the surface of the cone.

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## Procedure:

Step-1 Draw a horizontal x-y line of some suitable length.

Step-2 Draw a circle of radius 32.5 mm below the x-y line at some suitable distance from it. Divide this circle into 12 equal divisions as shown in the figure. Give notations on it. It is top view of the cone.

Step-3 From the center of the circle in the top view, draw a vertical center line of length 75 mm from the x-y line as shown in the figure. And from these notations of  the circle draw vertical projectors up to x-y line, and then converge all these projectors at the end of the center line i.e. apex of the cone (point o’) as shown into the figure. It is a triangular shape. And Give the notations on it.

Step-4 Draw a horizontal center line at the distance 25 mm above the x-y line, and a vertical center line at the distance 10 mm on the right of the vertical center line as shown in the figure. From the intersection of the above two center lines, draw a circle of radius 15 mm.

Step-5 Give the cutting points name at the intersection of the previously drawn circle with the vertical projectors of the circle in front view, i.e., 1’, 2’, 3’ etc. as shown into the figure.

Step-6 Now find out the angle covered by the cone when it is opened completely, by the equation , where L = Length of  the last generator i.e., o’-a’ or o’-g’.  = Angle subtended by the two extreme generators of the cone, when it is opened completely. r = Radius of the base circle of the cone. From this equation find out the value of  in degree.

Step-7 Draw a line parallel to and equal to the length of the last generator i.e., o’-g’, at some suitable distance from the front view. Give that line the name O-A as shown in the figure. With O as center and radius equal to OA, draw an arc such that the angle subtended by the arc should be equal to , which you have found out from the previous equation. This is the full development of the vertical surface of the cone.

Step-8 Divide this developed surface of the cone in 12 equal divisions, angle wise, as shown in the figure. And give the notations in capital letters. i.e., A, B, C etc.

Step-9 Now in the front view, from the cutting points on  the small circle of radius 15 mm, draw horizontal lines, parallel with x-y line, such that these lines should meet with the last generator i.e., o’-g’, in the front view.

Step-10 Then measure the distances of  the end points of previously drawn respective lines from the point o’, and transfer these distances in the developed surface of the cone on respective generators, as shown in the figure.

Step-11 Now Connects all the points in sequence with medium dark  smooth curve, as shown in the figure, and draw the boundary of the cone with dark curve with the use of a compass. This is the Development of the vertical surface of the cone.

Step-12 Give the dimensions by any one method of dimensions and give the notations as shown into the figure.