Problem 8.2 Development of Surfaces of Solids – The square pyramid with the length of side of base 30 mm and length of axis 60 mm as shown in the fig. 1 below. Develop the surface of the pyramid.
Step-1 Draw a horizontal x-y line of some suitable length.
Step-2 Draw a square of side of base 30 mm below the x-y line at some suitable distance from it, and in such a way that one of the sides of the square should be inclined at the angle 30° with the horizontal line, as shown in the figure. Then give notations on it, and from these notations draw lines meeting at the center of the square. It is top view of the square pyramid.
Step-3 From the center of the square in the top view, draw a vertical center line of length 60 mm from the x-y line as shown in the figure. And from all other notations of the square in the top view, 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 square pyramid (point o’) as shown into the figure. It is the front view of the square pyramid. And Give the notations on it.
Step-4 Mark a point at the distance 20 mm on the vertical center line, from the apex of the pyramid. Make this point as a center and draw an arc with the radius 10 mm between the last two edges of the pyramid. And mark cutting points at the intersections of the arc with the vertical edges i.e., 1’, 2’, 3’ etc.
Step-5 Draw a line parallel to and equal to the length of the last generator i.e., o’-c’, 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 of some suitable length, then cut this arc into four division of the length equal to the length of the side of the base which is 30 mm, as shown in the figure. Then give the notations on it and connect these points i.e., A,B,C,D,A in sequence with straight lines. This is the full development of the vertical surface of the square pyramid.
Step-6 From the cutting points in the front view i.e., 1’, 2’, 3’ etc. , draw horizontal lines up to the last generator i.e., o’-c’,
Step-7 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 pyramid on respective generators, which are 1,2,3,4,1, as shown in the figure.
Step-8 Now Connect all the cutting points i.e., 1,2,3,4,1 in sequence with medium dark smooth curve, as shown in the figure, and draw the generators which are below the smooth curve as medium dark straight lines. This is the Development of the vertical surface of the square pyramid.
Step-9 Give the dimensions by any one method of dimensions and give the notations as shown into the figure.