engineering graphics unit -3.docx

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Projections of Planes

Plane surface (plane/lamina/plate)

A plane is as two dimensional surface having length and breadth with negligible thickness. Theyare formed when any three non-collinear points are joined. Planes are bounded by straight/curvedlines and may be either regular or an irregular. Regular plane surface are in which all the sides

are eual. !rregular plane surface are in which the lengths of the sides are uneual.

Positioning of a Plane surface

A plane surface may be positioned in space with reference to the three principal planes of

projection in any of the following positions"

• Parallel to one of the principal planes and perpendicular to the other two.

• Perpendicular to one of the principal planes and inclined to the other two.

• !nclined to all the three principal planes.

Projections of a Plane surface

A plane surface when held parallel to a plane of projection# it will be perpendicular to the other

two planes of projection. The view of the plane surface projected on the plane of projection to

which it will be perpendicular will be a line# called the line view of a plane surface. $hen the plane surface is held with its surface parallel to one of the planes of projection# the view of the

plane surface projected on it will be in true shape because all the sides or the edges of the planesurface will be parallel to the plane of projection on which the plane surface is projected.$hen a plane surface is inclined to any plane of projection# the view of the plane surface

projected on it will be its apparent shape.

A few e%amples of projections of plane surfaces are illustrated below"

A: Plane surface parallel to one plane and perpendicular to the other two

&onsider A triangular lamina placed in the first uadrant with its surface parallel to 'P and

perpendicular to both (P and left PP. The lamina and its projections on the three projection

planes are shown in figure ).

a'b'c' is the front view# abc the top view and a’’b’’c’’ the side view

*ince the plane is parallel to VP # the front view a'b'c' shows the true shape of the lamina.

*ince the lamina is perpendicular to both (P and PP# the top view and side views are seen as

lines.



+igure ). Projections of a triangular lamina on the projection planes

After projecting the triangular lamina on 'P# (P and PP# both (P and PP are rotated about ,

and ,)) lines# as shown in figure # till they lie in-plane with that of 'P

+igure . Rotation of PP and (P after projection.



The orthographic projections of the plane# shown in figure can be obtained be the following

steps.

0raw , and ,)) lines and mark HP, VP and left PP . 0raw the triangle a'b'c' in true shape torepresent the front view at any convenient distance above the , line. !n the top view the

triangular lamina appears as a lineparallel to the , line. 1btain the top view acb as a line by

projecting from the front view at any convenient distance below the , line.

*ince the triangular lamina is also perpendicular to left PP# the right view will be a line parallel to

the ,)) line. To project the right view# draw a 234 line at the point of intersection of the , and

,)) lines.

0raw the hori5ontal projector through the corner a in the top view to cut the 234 line at m.

Through m draw a vertical projector. +rom the corners c6 and a6 in the front view draw the

hori5ontal projectors to cut the vertical projector drawn through m at c77 and b77. !n the right view

the corner A coincides with 8 and hence is invisible.

+igure . 1rthographic projections of the lamina A8&

B) Plane parallel to HP and perpendicular to both VP and PP

A suare lamina 9plane surface: is placed in the first uadrant with its surface parallel to (P

and perpendicular to both 'P and left PP. +igure 2 9a: shows the views of the object when



projected on to the three planes. Top view is shown as abcd # the front view as a’(d’)b’(c’) and

the side view as b”(a”)c”(d”). *ince the plane is parallel to the (P# its top view abcd will be in

its true shape. *ince the plane is perpendicular to 'P and PP# its front and side views will be lines

a’(d’)b’(c’) and b”(a”)c”(d”) respectively.

After projecting the suare lamina on 'P# (P and PP# both (P and PP are rotated about , and

,)) lines # as shown in figure 29b: # till they lie in-plane with that of 'P.

+igure 2. Projections of the lamina with its surface parallel to (1 and perpendicular to both 'P

and PP.

The orthographic projections of the plane# shown in figure 29c: can be obtained be the following

steps.

0raw , and ,)) lines and mark (P# 'P and left PP.0raw the suare abcd in true shape to represent the top view at any convenient distance below

the , line.

!n the front view# the suare lamina appears as a line parallel to the , line. 1btain the front

view as a line a'(d')b'(c') by projecting from the top view# parallel to the , line at any

convenient distance above it. !n the front view# the rear corners 0 and & coincide with the front

corners A and 8# hence d' and c' are indicated within brackets.



*ince the suare lamina is also perpendicular to left PP# the right view projected on it will also be

a line perpendicular to ,)) line. Project the right view as e%plained in the previous case. !n right

view# the corners A and 0 coincide with the corners 8 and & respectively# hence 9a6: and 9d6:# are

indicated within brackets.

C) Plane parallel to PP and perpendicular to both HP and VP

A pentagon lamina 9plane surface: is placed in the first uadrant with its surface is parallel to

left PP and perpendicular to both 'P and (P.

+igure 3 9a: shows the views of the object when projected on to the three planes. *ide view is

shown as a”b”c’’d”e”# the front view as b79c7:a79d7:e7 and the top view as a(b)e(c)d .*ince the

plane is parallel to the PP# its side view a”b”c’’d”e” will be in its true shape. *ince the plane is

perpendicular to 'P and (P# its front and side views will be projected as lines.

After projecting the pentagon lamina on 'P# (P and PP# both (P and PP are rotated about ,

and ,)) lines # as shown in figure 39b:# till they lie in-plane with that of 'P.

+igure 3 Projections of a pentagonal lamina with its surface parallel to PP and perpendicular to

(P and 'P.



The orthographic projections of the plane# shown in figure 39c: can be obtained be the following

steps. 0raw , and ,)) lines# and mark HP, VP and left PP .0raw the pentagon a”b”c”d”e”

in true shape to represent the side view at any convenient distance above the , line and left of

,)) line. The top and front views of the lamina appear as lines perpendicular to , line.

1btain the front view b’(c’)a’(d’)e’ as a line by projecting from the right view at any convenient

distance from the ,)) line. !n the front view# the rear corners 0 and & coincide with A and 8

respectively# hence d7 and c7 are indicated within brackets. The orthographic projections of the

Plane# shown in figure 29c: can be obtained be the following step. *ince the pentagon lamina is

also perpendicular to (P# the top view also appears as a line. Project the top view from the right

and front views.

) Plane surface perpendicular to one plane and inclined to the other two

). Plane inclined at ; to 'P and perpendicular to (P

0raw the projections of a triangular lamina 9plane surface: placed in the first uadrant

with its surface is inclined at f to 'P and perpendicular to the (P.

*ince the lamina is inclined to 'P# it is also inclined to left PP at 9<= - ;:.

The triangular lamina A8& is projected onto 'P# (P and left PP.a’b’c’ > is the front view projected on on 'P.

a”b”c” > is the right view projected on left PP.

*ince lamina is inclined to 'P and PP# front and side views are not in true shape.*ince lamina is perpendicular to (P# its top view is projected as a line acb



+igure ? 9c: shows the multiview drawing of the lamina.

+igure ?. The projections of the triangular lamina

Problem !:A regular pentagon lamina of = mm side rests on (P with its plane surface

vertical and inclined at == to 'P. 0raw its top and front views when one of its sides is

perpendicular to (P.

"olution: The projections The pentagonal lamina has its surface vertical 9i.e.# perpendicular to

(P: and inclined at == t o'P.*ince the lamina is inclined to 'P# initially it is assumed to be

parallel to 'P. !n this position one of the sides of the pentagon should be perpendicular to (P.

Therefore# draw a regular pentagon a6b6c6d6e6 in the 'P to represent the front view with its side a6e6

perpendicular to (P. *ince the lamina is perpendicular to (P# the top view will be a line#



a9e:b9d:c. Assume that edge a7 e7 perpendicular to (P in the final position. The top view of the

lamina is now rotated about a9e: such that the line is inclined at =4 to , line# as shown by

points a)#b)# c)# d)# and e) in the right bottom of +igure ). 0raw vertical projectors from points

a)#b)# c)# d)# and e). 0raw hori5ontal projectors from points a7# b7# c7# d7# and e7. The intersection

gives the respective positions of the points !n the +ront view. @oin a)7#b)7# c)7# d)7# and e)7 to

obtain the +ront view of the lamina.

+igure ). 1rthographic projections of the pentagonal lamina

Problem #$ 0raw the front view# top view and side view of a suare lamina. The surface

of the lamina is inclined at to (P and perpendicular to 'P.

"olution$ The thre views of the suare lamina is shown in figure . *ince the lamina is

perpendicular to 'P# its front view will be a line Ba79b7: c7 9d7:C having length as the true length

of the edge of the suare and inclined at to , line. The corners 8 and & coincide with A and

0 in the front view. *ince the lamina is inclined to (P at # it is also inclined to the left PP at 9<=-

:. The suare lamina is projected on to 'P# (P and left PP. 0raw vertical projectors from

points a7# b7# c7 and d7. 1n any position on these lines construct the rectangle a-b-c-d such that

length ab and cd are eual to the true length of the suare edge. The rectangle a-b-c-d is the top

view of the lamina. The side view of the lamina aD#bD#cD and dD can be obtained by drawing

projectors from points a7#b7#c7and d7 and a# b# c# and d.



+igure . The projection sof the suare lamina as mentioned in problem .

Problem %. 0raw the Top view and front view of a circular lamina if the surface of the lamina is

perpendicular to (P and inclined at =4 to 'P.

"olution: The projections of the circular lamina is shown in figure . Eet us first assume that the

plane is perpendicular to (P and parallel to 'P. The +ront view will be a circle and with

diameter eual to the diameter of the lamina. 0ivide the circle in to ) eual parts and label then

as 1’, 2’, 3’, …., 12’ . The top view will be a straight line 1-7 # parallel to , line and can be

obtained by drawing projectors from )7# 7# F. and )7. *ince the circle is inclined at =4 to 'P

and perpendicular to (P# reconstruct the top view such that the straight line is inclined at =4 to

, line. Eet the respective points be ))# )# )# F. )). 0raw vertical projectors from points

11, 21, 31, …. 121 to meet the hori5ontal projectors from points 1’, 2’, 3’, … 12’ to obtain the

points 11’, 21’, 31’, …. 121’ in the +ront view. 0raw a smooth curve passing through points

11’, 21’, 31’, …. 121’ to obtain the +ront view of the circular lamina.



+igure . Projections of the circular lamina mentioned in problem .

Projections of solids

"olid

A solid is a -0 object having length# breadth and thickness and bounded by surfaces which may

be either plane or curved# or combination of the two.

*olids are classified under two main headings

Polyhedron

*olids of revolution

A regular polyhedron is solid bounded only by plane surfaces 9faces:. !ts faces are formed by

regular polygons of same si5e and all dihedral angles are eual to one another. when faces of a polyhedron are not formed by eual identical faces# they may be classified into prisms and

pyramids.

+ive regular polyhedral are shown in figure )



+igure )" +ive regular polyhedra

Prism

Prisms are polyhedron formed by two eual parallel regular polygons# end faces connected by

side faces which are either rectangles or parallelograms.

0ifferent types of prisms are shown in figure



+igure . 'arious types of prisms generally encountered in engineering applications

"ome definitions regarding prisms

8ase and lateral faces" $hen the prism is placed vertically on one of its end faces# the end face

on which the prism rests is called the base. The vertical side faces are the lateral faces# as shownin +igure .

+igure . *hows the base and lateral face of a prism.

Base edge/"horter edge: These are the sides of the end faces# as shown in figure 2.

+igure 2. showing the base edge or shorter edge of a pentagonal prism.

A&is > it is the imaginary line connecting the end faces is called a%is and is shown in figure 3.



+igure 3 showing the A%is of a triangular prism.

'onger edge/lateral edges: These are the edges connecting the respective corners of the two end

faces. The longer edge of a suare prism is illustrated in figure ?.

+igure ?. illustrating the longer edge of a suare prism.

ight prism > A prism whose a%is is perpendicular to its end face is called as a right prism

.Prisms are named according to the shape of their end faces# i.e# if end faces are triangular# prism

is called a triangular prism.

bli*ue prism: !t is the prism in which the a%is is inclined to its base.

P+ramids

Pyramid is a polyhedron formed by a plane surface as its base and a number of triangles as its

side faces# all meeting at a point# called verte% or ape%.

A&is > the imaginary line connecting the ape% and the center of the base.



,nclined/slant faces > inclined triangular side faces.

,nclined/slant/longer edges > the edges which connect the ape% and the base corners.

ight p+ramid > when the a%is of the pyramid is perpendicular to its base.

bli*ue p+ramid > when the a%is of the pyramid is inclined to its base.

"olids of re-olution

when some of the plane figures are revolved about one of their sides > solids of revolution is

generated some of the solids of revolution are"

). &ylinder" when a rectangle is revolved about one of its sides# the other parallel sidegenerates a cylinder.

. &one" when a right triangle is revolved about one of its sides# the hypotenuse of the righttriangle generates a cone.

. 1bliue cylinder" when a parallelogram is revolved about one of its sides# the other parallel side generates a cylinder.

2. *phere" when a semi-circle is revolved about one of its diameter# a sphere is generated..

3. Truncated and frustums of solids > when prisms# pyramids# cylinders are cut by cutting

planes# the lower portion of the solids 9without their top portions: are called# eithertruncated or frustum of these solids. *ome e%amples are shown in figure G.



+igure G. !llustrates some e%amples of truncated / frustrum of solids.

Visibilit+

$hen drawing the orthographic views of an object# it will be reuired to show some of thehidden details as invisible. To distinguish the invisible portions from the visible ones# the

invisible edges of the object are shown on the orthographic views by dashed lines. (owever# in practice# these lines of dashes conveniently and collouially# but wrongly called as dotted lines.

To identify the invisible portions of the object# a careful imaginative thinking is essential.

ules of -isibilit+

$hen viewing an object# the outline of the object is visible. (ence the outlines of all the views

are shown by full lines. All the visible edges will be shown as solid lines as shown in figure H.+igure shows the frustum of a pentagonal pyramid.

+igure H. +ront view of the object. The visible edges are shown as solid lines and the hiddenedges are shown as dashed lines.

+igure < shows the projections of the object. !n the top view# the highest portions of the object

are visible. The top face A8&0I is at the top and is completely visible in the top view. !n the top

view# edges ab# bc# cd# de and ea are shown as full lines. The bottom pentagonal faces

A)8)&)0)I) is smaller than the top face# hence invisible. The slant edges AA)# 88)# &&)# 00)

and II) are invisible in the top view# hence they are shown as lines of dashes. The line

connecting a visible point and an invisible point is shown as an invisible line of dashes unless

they are out lines.

!n the front view# the front faces of the object are shown as visible. The faces A88)A) and

8&&)8) are the front faces. (ence in the front view# the corners a# b# c and a)# b)# c) are visible

to the observer. (ence in the front view# the lines a7a7)# b7b7) and c7c7) are shown as full lines.

(owever the corners d# e# d) and e) are invisible in the front view. The lines# e7e7)# d7d7) are

invisible# hence shown as dashed lines. The top rear edges a7e7# e7d7 and d7c7 coincide with the

top front visible edges a7b7 and b7c7.



+igure <. Projections of the frustrum of a pentagonal pyramid.

!n the side view - the face lying on that side are visible. As seen in the left side view# the

corners e# a# b and e)# a)# b) lie on left side and are visible in the left view. (ence the lines# eDe)D#

aDa)D and bDb)D are shown as full lines. The edges dDd)D# cDc)D coincide with the visible edges

eDe)D and aDa)D respectively.

Projections of solids placed in different positions

The solids may be placed on (P in various positions

). The way the a%is of the solid is held with respect to (P or 'P or both -

o Perpendicular to (P or 'P

o Parallel to either (P or 'P and inclined to the other



o !nclined to both (P and 'P

. The portion of the solid on which it lies on (P# e%cept when it is freely suspended

position. !t can lie on (P on its base edge or a corner# or a lateral face# or ape%.

A&is of the solid perpendicular to HP

A solid when placed on (P with its a%is perpendicular to it# then it will have its base on (P. This

is the simplest position in which a solid can be placed. $hen the solid is placed with the base on(P position# in the top view# the base will be projected in its true shape. (ence# when the base of

the solid is on (P# the top view is drawn first and then the front view and the side views are

projected from it. +igure )= shows a cylinder with its a%is perpendicular to (P. There is only one position in which a cylinder or a cone may be placed with its base on (P.

+igure )=. +ront view and top view of a cylinder and cone

+or prisms# there are 2 positions it may be placed with its base on (P.These positions are

illustrated in figure )).



+igure )). Projections of a triangular prism resting on its base on (P with different positions.

There are 2 positions in which pyramids may be placed with its base on (P. These positions are

shown in figure ).

+igure ). Projections of a triangular Pyramid resting on its base on (P with different positions.



Projections of a solid with the a&is perpendicular to VP

$hen a solid is placed with its a%is perpendicular to 'P# the base of the solid will always be

perpendicular to (P and parallel to 'P. (ence in the front view# base will be projected in true

shape. Therefore# when the a%is of the solid is perpendicular to 'P# the front view is drawn first

and then the top and side views are drawn from it. $hen a cylinder rests on (P with its a%is

perpendicular to 'P# one of its generators will be on (P.

+igure ) shows the +ront view and Top view of a cylinder and cone resting on (P with their

a%es perpendicular to 'P. !n this case one of the points on the circumference of the base will be

on ,.

+igure ) shows the +ront view and Top view of a cylinder and cone

Prism may be placed with their a%is perpendicular to 'P in three different positions. The

different positions are shown in figure )2.



+igure )2. Projections of a pentagonal prism resting on (P and a%is perpendicular to 'Pwith

different positions.

As shown in +igure )3# pyramid may be placed with their a%is perpendicular to 'P in threedifferent positions.



+igure )3. Projections of a pentagonal pyramid resting on (P and a%is perpendicular to 'P with

different positions.

A&is of the solid inclined to HP and parallel to VP

$hen a solid is placed on (P with its a%is inclined to (P# the elemental portion of the solid that

lies on (P depends upon the type of the solid.

$hen a prism is placed on (P with its a%is inclined to it# then it will lie either on one of its base

edges or on one of its corners on (P.

$hen a pyramid is placed on (P with its a%is inclined to (P# then we will have one of its base

edges on (P or one of its base corners on (P or one of its slant edges on (P or one of its

triangular faces on (P or an ape% on (P.

.ethods of drawing the projections of solids

These are two methods for drawing the projections of solids"

). &hange of position method.

. Au%iliary plane method 9&hange of reference-line method:

Change of position method

!n this method the solids are placed first in the simple position and then tilted successively in

two or three stages to obtain the final position. The following are some of the e%amples.



a$ .ethod of obtaining the top and the front -iews of the p+ramid when it lies on HP on one

of its base edges with its a&is or the base inclined to HP$

!f the solid is reuired to be placed with an edge of the base on (P# then initially the solid has to be placed with its base on (P such that an edge of the base is perpendicular to 'P# i.e.# to ,

line in top view preferably to lie on sthe right side.$hen a pentagonal prism has to be placed with an edge of base on (P such that the base or a%is

is inclined to (P# then initially# the prism is placed with its base on (P with an edge of the base perpendicular to 'P and the lying on the right side. !n this position# the first set of top and front

views are drawn with the base edges 9c):9d): perpendicular to , line in the top view. !n the

front view# this edge c)79d)7: appears as a point.*ince the prism has to lie with an edge of the base on (P# the front view of the prism is tilted on

the edge c)79d)7: such that the a%is is inclined at to (P.

Redraw the first front view in the tilted position. $henever the inclination of a%is with (P isgiven# first the base is drawn at 9<=- : in the front view# otherwise improper selection of the

position of the a%is may result in the base edge c)79d)7: lying above or below the , line. The

second top view is projected by drawing the vertical projectors from the corners of the secondfront view and the hori5ontal projectors from the first top view. +igure ) shows the seuence in

obtaining the projection of the solid for the above case.

+igure ). !llustrating the seuence for obtaining the projections of a pentagonal prism placed

with an edge of base on (P such that the base or a%is is inclined to (P



b.Corner of the base on HP with two base edges containing the corner on which it rests

ma0e e*ual inclinations with HP

$hen a solid lies on one of its corners of the base on (P# then the two edges of the base

containing the corner on which it lies make either eual inclinations or different inclination with

(P. !nitially the solid should be placed with its base on (P such that an imaginary line

connecting the center of the base and one of its corners is parallel to 'P# i.e. to , line in the top

view# and preferably to lie on the right side. +or e%ample# when a he%agonal prism has to be

placed with a corner of the base on (P such that the base or the a%is is inclined to (P# then

initially the prism is placed with its base on (P such that an imaginary line connecting the center

of the base and a corner is parallel to 'P and it lies on the right side. !n this position# the first set

of top and front views are drawn # as shown in step-) of figure . The line 9o):9d): is parallel to

the , line in the top view.



+igure " Projections of a prism with a corner of the base on (P and the a%is is inclined to (P.

*ince the prism has to lie on one of its corners of the base on (P# the front view of the prism is

tilted on the corner d)7 such that the a%is is inclined at to (P. Redraw the front view in the

tilted position as shown in *tep- of figure-. The base edge is drawn at 9<=- : in the front view.

The second top view is projected by drawing the vertical projectors from the corners of the

second front view and hori5ontal projectors from the first top view.

+ollowing the above procedure# the top and front views of the pyramid when it rests on (P on

one of its base corners such that the two base edges containing the corner on which it rests make

eual inclinations with (P is shown in figure .

+igure showing the projection of a pyramid resting on (P on one of its base corners with two

base edges containing the corner on which it rests make eual inclinations with (P

c$ Projections of a p+ramid l+ing on one of its triangular faces on HP



!f a pyramid has to be placed on one of its triangular faces on (P# then initially let the pyramid

be placed with its base on (P such that the base edge containing that face is perpendicular to 'P.

9i.e. perpendicular to , line:. +igure 2 illustrates the seuence in obtaining the projections of

the pyramid. !n the first front view# the inclined line# o1c1(d1) represents a triangular face.

Redraw the front view such that the line representing the triangular face o1c1(d1) lies on (P.

Project the top view in this position.

+igure 2 !llustrates the seuence in obtaining the projections a pyramid lying on one of its

triangular faces on (P

d$ Projections of a p+ramid l+ing on one of its slant edge HP

The seuence of obtaining the projections of a pyramid lying on one of its slant edge on (P is

shown in figure 3. !n step-)# The +' and T' of the pyramid in the simple projection is drawn

such that in the# top view the slant edge 9line cd: on which it will rest on ground is drawn

parallel to (P 9 parallel to , line: in the. !n the front view this edge will be line c7 o7. !n step-



# the object is then rotated such that the pyramid lies with its edge o7c7 on (P. i.e. in the front

view# o7c7 lies on , line. Project the Top view from this +ront view.

+igure 3. Projections of a pyramid lying on one of its slant edges on (P



Problem !$

A cube of = mm sides is held on one of its corners on (P such that the bottom suare facecontaining that corner is inclined at == to (P. Two of its adjacent base edges containing the

corner on which it rests are eually inclined to 'P. 0raw the top and front views of the cube.

"olution:

The procedure of obtaining the projections is shown in figure ?. !n*tep-)# the projections of the

cube is drawn in the simple position. The cube is assumed to lie with one of its faces completely

on (P such that two vertical faces make eual inclinations with 'P. 0raw a suare abcd to

represent the top view of the cube such that two of its sides make eual inclinations with the ,

line# i.e.# with 'P. Eet 9a):# 9b):# 9c): and 9d): be the four corners of the bottom face of the cube

which coincide in the top view with the corners a# b# c and d of the top face. Project the front

view of the cube. The bottom face a)7b)7c)79d)7: in the front view coincide with the , line.

Jow the cube is tilted on the bottom right corner c)7 9step-: such that the bottom face

a)7b)7c)79d)7: is inclined at == to (P. Reproduce the front view with face a)7b)7c)79d)7: inclined

at == to the , line.

0raw the vertical projectors through all the corners in the reproduced front view and hori5ontal

projectors through the corners of the first top view. These projectors intersect each other to give



the corresponding corners in the top view

+igure ). The projections of the cube of problem ).

Problem2#$

A cube of = mm side rests with one of its edges on (P such that one of the suare faces

containing that edge is inclined at == to (P and the edge on which it rests being inclined to ?==

to 'P. 0raw its projections.

"olution$The procedure of obtaining the projections is shown in figure G. +irst the T' and +' of the cube

is drawn with the cube in the simple position. The edge bc is drawn perpendicular to the , line.

!n step# the cube is tilted such that the base of the cube is inclined at == to (P. The front view is

reproduced with b)7 c)7a)7d)7 inclined at == to ,. The top view of the cube in step- is

obtained by drawing projectors mentioned in problem ). !n step-# the top view in step- is



rotated such that line c) b) is inclined at ?= = to , line. The front view in step- is obtained by

drawing projectors from the top view in step- and +ront view in *tep-.

+igure . The projections of the cube of problem .

Problem %

An euilateral triangular prism = mm side of base and 3= mm long rests with one of its shorter

edges on (P such that the rectangular face containing the edge on which the prism rests is

inclined at == to (P. The edge on which prism rests is inclined at ?== to 'P. 0raw its

projections.

"olution: The procedure of obtaining the projections is shown in figure H. The prism rests with

one of its shorter edges# i.e.# triangular or base edge on (P such that the rectangular face

containing that edge is inclined at == to (P.

0raw the simple views of the prism when it rests with one of its triangular faces# i.e.# base



completely lying on (P and also with one of its shorter edges perpendicular to 'P# i.e.# to ,

line. The shorter edge 9b):9c): is perpendicular to the , line. The rectangular face containing

the edge b)79c)7: is b)7b79c7:9c)7:.

Jow tilt the prism on the edge b)79c)7: such that the rectangular face b)7b79c7:9c)7: is inclined at

== to the , line. !n this tilted position# project the top view.

!t is seen that the edge b)c) in the top view shown is perpendicular to 'P# i.e# to , line. 8ut the

edge b)c) has to be inclined at ?== to 'P# i.e# to , line. Therefore# reproduce the top view with

the edge b)c) inclined at ?== to the , line as shown in the top view.

Project the reproduced top view to get the front view.

+igure . The projections of the triangular prism of problem-

Problem23A he%agonal pyramid has an altitude of ?= mm and side base =mm. The pyramid rests on one

of its side of the base on (P such that the triangular face containing that side is perpendicular to

(P. 0raw the front and top views.

"olution: The solution the problem is shown in figure <. !n step-)# the pyramid is drawn in the



simple position with base edge cd perpendicular to , line. !n *tep-# the +ront view is tilted

about cd such that line o7c7d7 is made perpendicular to , line. The top view is obtained by

drawing projectors from the top view of step ) and front view in step-.

+igure 2. The projections of the he%agonal pyramid of problem-2.

Problem24

Problem 0raw the top and front views of a rectangular pyramid of sides of base 2=% 3= mm and

height G= mm when it lies on one of its larger triangular faces on (P. The longer edge of the



base of the triangular face lying on (P is inclined at ?== to 'P in the top view with the ape% of

the pyramid being nearer to 'P.

"olution :

The solution the problem is shown in figure 3. The projectors are obtained in -steps as

illustrated in the figure. !n the first step# the solid is projected in the simple position with base

8& perpendicular to 'P. !n the second step# the solid is tilted about the edge 8& such that the

face 8&1 is made to lie on the ground. The front view is rotated and the top view is projected

from the front view and the top view in the first step. !n step-# the top view is rotated such that

edge 8& is inclined at ?=4 to , line . The +ront view is projected using this top view and +ront

view of *tep-.

+igure 3. The projections of the rectangular pyramid of problem-3.



Problem25

A cone of base H= mm diameter and height )== mm lies with one of its generators on (P and the

a%is appears to be inclined to 'P at an angle of 2== in the top view. 0raw its top and front views.

"olution:

+igure ? illustrates the procedure for obtaining the projections of the cone. Three steps are

involved. !n step-)# the Top view and +ront 'iew of the cone is drawn in the simple position.

The base circle is divided in to ) eual parts. These points ate joined with the ape% to obtain the

respective generators. !nstep # the cone is tilted such that the cone lies on one of its generator in

the (P. i.e. the generator g7o7 is made to coincide with the , line. The top view of the object in

this condition is drawn by drawing projectors. !n step-# the cone is titled such that in the top

view the a%is is inclined at 2== to the , line. The front view of the object is obtained by

projection techniue.

+igure ?. The projections of the cone of problem-?.



Problem26

0raw the top and the front views of a right circular cylinder of base 23 mm diameter and ?= mm

long when it lies on (P such that its a%is is inclined at 3= to (P and the a%is appears to be

perpendicular to 'P in the top view.

"olution:

The solution to the problem is illustrated in figure-G . Three steps are involved as shown in the

figure. !n *tep-)# the cylinder is drawn in the simple position 9resting on the base on (P:. The

circle in the top view is dived in to ) eual parts and then projected in to the front view. !n

step-# The +ront view is rotated about g) such that the a%is is inclined at 34 to (P 9or ,line:.

The top view is projected from this front view with thehelp of Top view in step-). . !n step # the

top view is rotated such that a%is is perpendicular to , line. The front view is then projected

from the top view.



+igure G. The projections of the cylinder as per problem-G.

engineering graphics unit -3.docx

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