Development of Pyramid

DEVELOPMENT OF PYRAMID

Development of lateral surface of pyramid is drawn by radial line method in a similar way as that of cone. Since the pyramid has regular edges its lateral development is a series of isosceles triangle representing the true side of the base and true slant length. The true side of the base may be directly taken from the top view for any position of pyramid since the axis of pyramid is perpendicular to HP, where as the true slant edge of pyramid is to be taken from the front view.

Consider a square pyramid resting on its base on HP, two adjacent sides are equally inclined to XY as shown in Fig. 16.40 (a). True length of side (ie., ab) may be taken from the top view. The top view of end generator o'a' is (ie., oa) is parallel to XY. Hence, o'a' gives the true slant length.

If the square pyramid is placed in such a way that its two faces are parallel to XY as shown in Fig. 16.40 (b) the top view of end generator o'b' (ie., ob) is not parallel to XY. Hence o'b' will not represent the true slant length. To find out the true slant length in this case, with o as centre and ob as radius draw an arc in top view to cut the horizontal line passing through o at b1. Then project the point b1, to front view as b1'. Join o' and b1'. The line o' b1' represents the true slant length.

After finding the true slant length for the given position of base of pyramid draw an arc of radius equal to true slant length and step-off the arc into required number of divisions equal to the number of sides of base and then join with O to get series of isosceles triangles, which is the required development of the given pyramid.

The development of pyramid with cut out / hole can be developed in similar way as that of other solids. Draw the development of full pyramid first and then transfer the cut out / hole to the development from front view.

Example 34 : Draw the development of lateral surface of a hexagonal pyramid with a base side 30 mm and axis 60 mm long which is resting on its base on HP such that two sides of base are parallel to VP.

Step 1 (Top view): Draw the top view as a hexagon of side 30 mm in such a way that two sides are parallel to XY. Name the corners as a, b, c etc.

Step 2 (Front view): Project the top view and complete the front view. Name the points as a', b' etc. accordingly. Top view of end generators o'a' (or o'd') are parallel to XY. Hence length of o'a' (or o'd') represents the true slant length. Measure the length o'a'.

Step 3 (Development) :

1. Mark a point O and with O as centre and radius equal to the true slant length (ie., o'a' (or) o'd') draw an arc.

2. Draw vertical line through O and locate the point A on the arc such that OA = o'a'.

3. Starting from the point A, mark 6 equal divisions (equal to the number of sides of base of the pyramid) and name the points as B, C, D, E, F and A such that AB = BC = CD = DE = EF = FA = 30 mm and join these points with straightlines.

4. Join the points A, B, C etc. with O, to get series of isosceles triangles representing the triangular faces of pyramid, which is the required development of the pyramid.

Example 35 : In the above problem draw the development of lateral surface if the hexagonal pyramid is placed in such a way that two edges of the base are perpendicular to VP.

1. Draw the top view and front view.

2. Here the top view of end generator o'a' (ie., oa) is not parallel to XY. Hence o'a' will not represent the true slant length.

3. With o as centre and oa as radius draw an arc in top view to cut the horizontal line passing through o at a1. Transfer the point a1 to front view as a1'.

4. Join o' and a1' which represent the true slant length.

5. Draw the development of lateral surface as explained in previous problem, by taking the radius of arc as o' a1'.

Example 36: A regular hexagonal pyramid side of base 30 mm and axis 60 mm long is resting on its base on HP such that two of the sides of the base are perpendicular to VP. It is cut by a plane inclined at 40° to HP and perpendicular to VP, bisects the axis of the pyramid. Draw the development of the lateral surface of the truncated pyramid.

1. Draw the top view as a hexagon of side 30 mm, keeping two of the sides of base are perpendicular to XY.

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

3. Draw the cutting plane at an angle of 40° to HP and bisecting the axis of pyramid.

4. Top view of end generator (ie oa) is not parallel to XY. Hence o'a' will not represent the true slant length. Hence, with o as centre and oa as radius draw an arc to cut the horizontal line drawn through o at a1.

5. Transfer the point a1 to front view as a1'. Now o'a1'a represents the true slant length.

6. Mark the points of intersection of cutting plane with the vertical edges of pyramid as p'(u'), q'(t') and r'(s').

7. Draw the horizontal line through these points to cut the true slant length o'a1' at P1', q1' and r1'.

8. Mark the point O. With O as centre and o'a1' as radius draw an arc.

9. Mark the point A on the arc and other points B, C, D, E, F and A such that AB = BC = CD = DE = EF = FA = 30 mm.

10. Join these points and then connect with O which represents the development of lateral surface of full pyramid.

11. On the development mark P on OA such that OP = o'p1'.

12. Similarly mark the points Q, R, S, T and U on the respective generators.

13. Join the points P, Q, R, S, T, U, P using thick lines. Now the shape obtained is the required development of given truncated pyramid.

Example 37: A pentagonal pyramid, side of base 30 mm and height 55 mm is resting on HP with its base and an edge of the base is parallel to VP and neater to it. It is cut by a plane perpendicular to VP, and inclined at 40° to HP, passing through a point on the axis, 32 mm above the base. Draw the sectional top view and development of the lateral surface of the truncated pyramid.

The sectional top view and development are shown in Fig. 16.44.

Example 38: Draw the development of lateral surface of a hexagonal pyramid with a 40 mm base side and 60 mm long axis, resting on its base on HP such that an edge of the base is perpendicular to VP and cut by a horizontal section plane at a height of 25 mm from the base.

The development is shown in Fig. 16.45.

Example 39: A square pyramid of base 35 mm side and 70 mm long is resting on HP such that two adjacent sides of the base are equally inclined to VP. It is sectioned by a plane perpendicular to VP inclined at 30° to HP and Passing through the midpoint of the axis. Draw the sctional top view and development of lateral surface of the truncated Pyramid.

The development is shown in Fig. 16.46.

Example 40: A square pyramid of base side 30mm and axis 60mm rests on its base on the ground with one of the base edge inclined at 30° to the VP. A string is wound round the surfaces of the pyramid starting from left extreme point on the base and ending at the same point. Find the shortest length of the string required. Also trace the path of the string in the front view and top view.

1. Draw the top view as a square of side 30mm keeping a side inclined at 30° to VP. Name the corners as a, b, c and d.

2. Project the top view and draw front view.

3. Draw the development of the lateral surface of the pyramid with true slant height 1, consisting of 4 triangles of base 30mm each. Name the points as A, B, C and D.

4. Join the extreme points AA on the development by straight line which gives the shortest length of the string and mark the intersecting points as 1, 2 and 3 with the lines OB, OC and OD respectively.

5. Mark the points 1', 2′ and 3′ in front view on the respective generators (For example mark 1' on the generator o'b' such that 0-1 in development = o'b' in front view). Similarly mark the points 2' and 3'.

6. Join the points a', 1', 2' 3' and a' in front view (note that the line joining the points 2' 3' and P' is shown in dashed line since it is seen on the rear side of the pyramid.

7. Transfer the points a', 1', 2' and 3' to top view and mark as a, 1, 2 and 3 (for example draw a vertical projector through the point 1' to cut the line o-b in top view at 1). Join the points a, 1, 2, 3 and a by a straight line in top view which shows the path of the string.

1. Development of pyramid with cut-out

Example 41: The front view of a truncated square pyramid as shown in Fig. 16.48, having a base with a 40 mm side and a 65 mm long axis, is resting on its base on HP with all the sides of the base equally inclined to VP. Draw the development of the lateral surface of the retained solid.

The development of the truncated pyramid is shown in Fig. 16.49.

1. Draw the top view as a square, keeping all the sides of the base equally inclined to XY. (ie., at 45° to XY). Name the corners as a, b, c and d.

2. Project the top view and complete the front view. Name the points as a', b', c' and d'. 

(Note that the top view of end generator o'a' (ie. oa) is parallel to XY. Hence o'a' gives the true slant length).

3. Draw the cutting plane in front view and project the same to the top view.

4. Name the intersecting points in both front view and top view.

5. Draw the development of uncut pyramid in usual method.

6. With O as centre draw arcs 1-1 with 0'1' as radius and another arc with o'q1' as radius (q1' is the point of intersection of horizontal line drawn through q' with the end generator)

7. Similarly transfer the points of intersection from front view and top view on the corresponding generators to obtain the points P, Q, R, S, T.

8. Join these points in proper sequence and darken the edges of the development.

2. Development of Pyramid with hole

Example 42: A hexagonal pyramid with a 30 mm side and 70 mm long axis rests on the ground with a side of base parallel to VP. A circular hole with a 30 mm diameter is cut through the faces of the pyramid such that axis of the hole and pyramid intersects at right angle and 25 mm above the base. Draw the development of the lateral surface of the pyramid.

The development is shown in fig. 16.50.

Example 43: A pentagonal pyramid of base side 30mm and axis 60mm length rests on its base on HP with a base edge parallel to VP and farther away from VP. A square slot of diagonal 20mm is drilled right through the pyramid such that one diagonal coincides with the axis of the pyramid. The centre of the slot lies on the axis of the pyramid 25mm above the base of the pyramid. Draw the development of the lateral surfaces of the pyramid with the cut-out when the axis of the slot is perpendicular to VP.

The development is shown in Fig. 16.51.