Projections of Solids Mohammed Umair Hamid Assistant Professor

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Projections of Solids Mohammed Umair Hamid Assistant Professor (NSAKCET) Mechanical Department Engineering Graphics 1

Projections of Solids Mohammed Umair Hamid Assistant Professor (NSAKCET) Mechanical Department Engineering Graphics 1 Year 1 Semester

PROJECTIONS OF SOLIDS Definition of Solid: A solid is a three dimensional object having

PROJECTIONS OF SOLIDS Definition of Solid: A solid is a three dimensional object having length, breadth and thickness and it is completely bounded by either plane or curved, or combination of the two, is called as 3 -Dimensional Solid. -The shape of the solid is described by drawing its two orthographic views usually on the two principle planes i. e. H. P. & V. P. -For some complicated solids, in addition to the above principle views, side view is also required. -A solid is an aggregate of points, lines and planes and all problems on projections of solids would resolve themselves into projections of points, lines and planes.

Classification of Solids: Solids may be divided into two main groups (A) Polyhedra A

Classification of Solids: Solids may be divided into two main groups (A) Polyhedra A Polyhedra is defined as a solid bounded by planes called faces which meet in straight lines called edges. (B) Solids of revolution When a solid is generated by revolutions of a plane figure about a fixed line (Axis) then such solids are named as solids of revolution.

SOLIDS Polyhedra Solids of revolution (1) Prism (1) Cylinder (2) Pyramid (2) Cone (3)

SOLIDS Polyhedra Solids of revolution (1) Prism (1) Cylinder (2) Pyramid (2) Cone (3) Tetrahedron (3) Sphere (4) Cube or Hexahedron (5) Octahedron (4) Ellipsoid (8 Equilateral Triangles) (6) Dodecahedron (5) Paraboloid (12 Pentagons) (7) Icosahedron (20 Triangles) (6) Hyperboloid

There are seven regular Polyhedra which may be defined as stated above Solids of

There are seven regular Polyhedra which may be defined as stated above Solids of revolutions may be of following types

(1) Prism: It is a polyhedra having two equal and similar faces called its

(1) Prism: It is a polyhedra having two equal and similar faces called its ends or bases, parallel to each other and joined by other faces which are rectangles. The imaginary line joining the Centres of the bases or faces is called Axis of Prism. Edge Axis Faces

According to the shape of its base, prism can be sub classified into following

According to the shape of its base, prism can be sub classified into following types: (a) Triangular Prism (b) Square Prism (c) Pentagonal Prism (d) Hexagonal Prism

(2) Pyramid: This is a polyhedra having plane surface as a base and a

(2) Pyramid: This is a polyhedra having plane surface as a base and a number of triangular faces meeting at a point called the Vertex or Apex. The imaginary line joining the Apex with the Centre of the base is called Axis of pyramid. Edge Axis Base

According to the shape of its base, pyramid can be sub classified into following

According to the shape of its base, pyramid can be sub classified into following types: (a) Triangular Pyramid: (b) Square Pyramid: (c) Pentagonal Pyramid: (d) Hexagonal Pyramid:

Rectangle Axis Base (1) Cylinder: A right regular cylinder is a solid generated by

Rectangle Axis Base (1) Cylinder: A right regular cylinder is a solid generated by the revolution of a rectangle about its vertical side which remains fixed.

Right angle triangle Generators Axis Base (2) Cone: A right circular cone is a

Right angle triangle Generators Axis Base (2) Cone: A right circular cone is a solid generated by the revolution of a right angle triangle about its vertical side which remains fixed.

Important Terms Used in Projections of Solids: (1) Edge or generator: For Pyramids &

Important Terms Used in Projections of Solids: (1) Edge or generator: For Pyramids & Prisms, edges are the lines separating the triangular faces or rectangular faces from each other. For Cylinder, generators are the straight lines joining different points on the circumference of the bases with each other

Important Terms Used in Projections of Solids: (2) Apex of solids: For Cone and

Important Terms Used in Projections of Solids: (2) Apex of solids: For Cone and Pyramids, Apex is the point where all the generators or the edges meet. Apex Edges PYRAMID Generators CONE Apex

Rectangle Axis Generators Base Edge CYLINDER PRISM Axis Faces

Rectangle Axis Generators Base Edge CYLINDER PRISM Axis Faces

Important Terms Used in Projections of Solids: (3) Axis of Solid: For Cone and

Important Terms Used in Projections of Solids: (3) Axis of Solid: For Cone and Pyramids, Axis is an imaginary line joining centre of the base to the Apex. For Cylinder and Prism, Axis is an imaginary line joining centres of ends or bases.

(4) Right Solid: A solid is said to be a Right Solid if its

(4) Right Solid: A solid is said to be a Right Solid if its axis is perpendicular to its base. (5) Oblique Solid: A solid is said to be a Oblique Solid if its axis is inclined at an angle other than 90° to its base. Axis Base

(6) Regular Solid: A solid is said to be a Regular Solid if all

(6) Regular Solid: A solid is said to be a Regular Solid if all the edges of the base or the end faces of a solid are equal in length and form regular plane figures

(7) Frustum of Solid: CUTTING PLANE PARALLEL TO BASE When a Pyramid or a

(7) Frustum of Solid: CUTTING PLANE PARALLEL TO BASE When a Pyramid or a Cone is cut by a Plane parallel to its base, thus removing the top portion, the remaining lower portion is called its frustum. FRUSTUM OF A PYRAMID

Important Terms Used in Projections of Solids: (8) Truncated Solid : When a Pyramid

Important Terms Used in Projections of Solids: (8) Truncated Solid : When a Pyramid or a Cone is cut by a Plane inclined to its base, thus removing the top portion, the remaining lower portion is said to be truncated.

Class A(1): Axis perpendicular to H. P. and hence parallel to both V. P.

Class A(1): Axis perpendicular to H. P. and hence parallel to both V. P. & P. P. o’ Axis X a’, b’ c’, d’ a b d o c Y

Class A(2): Axis perpendicular to V. P. and hence parallel to both H. P.

Class A(2): Axis perpendicular to V. P. and hence parallel to both H. P. & P. P. f’, 6’ a’, 1’ e’, 5’ d’, 4’ b’, 2’ c’, 3’ 1 2, 6 3, 5 4 H X a b, f c, e d Y

Class A(3): Axis perpendicular to P. P. and hence parallel to both H. P.

Class A(3): Axis perpendicular to P. P. and hence parallel to both H. P. & V. P. L c’ a’, b’ 3’ 1’ 2’ a” 1” X a 1 c 3 b 2 c” 3” b” 2” Y

PROBLEM NO: 1 Draw the projections of a triangular prism, base 40 mm side

PROBLEM NO: 1 Draw the projections of a triangular prism, base 40 mm side and axis 50 mm long, resting on one of its bases on the ground with a vertical face perpendicular to the V. P. a’(b’) c’ O’ 50 X 1’ ( 2’) O’ 3’ b(2) c(3) O a(1) 40 Y

PROBLEM NO: 2 Draw the projections of a pentagonal pyramid, base 30 mm edge

PROBLEM NO: 2 Draw the projections of a pentagonal pyramid, base 30 mm edge and axis 50 mm long, having its base on the ground an edge of the base parallel to V. P. 50 o‘ X a’ e‘ b’ e d‘ e‘ d c o a b 30 Y

PROBLEM NO: 3 Draw the projections of i) a cylinder, base 40 mm dia

PROBLEM NO: 3 Draw the projections of i) a cylinder, base 40 mm dia and axis 50 mm long, ii) a cone, base 40 mm dia and axis 50 mm long, resting on the ground on their respective bases. o‘ c’ 50 b’ d’ 50 a’ X 2’ 4’ 1’ 3’ d Y X a’ b’ d’ c’ d 4 a 1 o 3 c a o 2 b b 40 40 c Y

PROBLEM NO : 4 A cube of 50 mm long edges is resting on

PROBLEM NO : 4 A cube of 50 mm long edges is resting on the H. P with its vertical faces equally inclined to the V. P. Draw its projections. b’(d‘) c’ 50 a’ 45⁰ 2’(4’) d(4) 3’ c(3) a(1) 50 X 1’ b(2) Y

PROBLEM NO: 5 A hexagonal prism has one of its rectangular faces parallel to

PROBLEM NO: 5 A hexagonal prism has one of its rectangular faces parallel to the ground, its axis is perpendicular to the VP and 3. 5 cm above the ground. Draw its projections when the nearer end is 2 cm in front of VP. Side of the base is 2. 5 cm, axis 5 cm long. 2. 5 f (6) e (5) d (4) a (1) b (2) c (3) 3. 5 X 1 2 (6) 5 (3) 4 2 50 a b (f) c (e) d Y

PROBLEM NO : 6. A hexagonal pyramid, base on the ground a side of

PROBLEM NO : 6. A hexagonal pyramid, base on the ground a side of the base parallel to and 25 mm in front of the VP. Draw the projections taking a side of the base 40 mm and axis 65 mm long. 65 O’ b’ (f’) c’ (e’) f a Y e d O b d’ 25 X a’ 40 c

PROBLEM NO: 8. A triangular pyramid base on the ground an edge of the

PROBLEM NO: 8. A triangular pyramid base on the ground an edge of the base inclined at 450 to the VP, the apex 40 mm in front of the VP. Draw the projections taking a side of the base 40 mm long and axis 65 mm long. 65 o’ o’ 45⁰ c’ b’ c 40 X a’ o a 40 b Y

PROBLEM NO: 9 A cylinder, axis perpendicular to the VP and 40 mm above

PROBLEM NO: 9 A cylinder, axis perpendicular to the VP and 40 mm above HP, one end 20 mm in front of the VP. Draw the projections taking diameter of the base 50 mm and the axis 65 mm long. (Home work). ⌽ 50 4 d’’ 3’ c’ 40 1’ a’ 20 a Y b d c 2 4 3 65 X 2’ b’ 1

PROBLEM NO: 10 A pentagonal prism, rectangular face parallel to and 10 mm above

PROBLEM NO: 10 A pentagonal prism, rectangular face parallel to and 10 mm above the HP, axis perpendicular to the VP and one base in the VP. Draw the projections taking side of the base 40 mm long and axis 65 mm long. e’ 2’ a’ 1’ 3’ d’ o 4’ c’ 5’ X 10 b’ 1 2 a b 40 5 3 4 65 e c d Y

PROBLEM NO: 11 A square pyramid, all edges of the base equally inclined to

PROBLEM NO: 11 A square pyramid, all edges of the base equally inclined to the HP and the axis parallel to and 50 mm away from both HP and VP. Draw the projections taking a side of the base 40 mm and axis 65 mm long. 65 d’’ a’(c’) o’’ c’’ 50 o’ 40 d’ b’ 45⁰ X b’’ Y a o d(b) c 50 45⁰

PROBLEM NO: 12 A cone, apex in the HP. Axis vertical and 40 mm

PROBLEM NO: 12 A cone, apex in the HP. Axis vertical and 40 mm in front of VP. Draw its projections taking dia of the base 50 mm and axis 65 mm long. (Home work). b’ d’ c‘ 65 a’ o‘ X Y a o 40 d c b ⌽ 50

PROBLEM NO: 13 A pentagonal pyramid, base in the VP and an edge of

PROBLEM NO: 13 A pentagonal pyramid, base in the VP and an edge of the base in the HP. Draw its projections taking a side of the base 40 mm and axis 65 mm long. e’ a’ X a 40 o’ d’ c’ b’ b e c Y d 65 o

INCLINED TO ONE PLANE PROBLEM NO: 1 Draw the projections of a pentagonal prism,

INCLINED TO ONE PLANE PROBLEM NO: 1 Draw the projections of a pentagonal prism, base 25 mm side and axis 50 mm long, resting on one of its rectangular faces on the ground with the axis inclined at 450 to the VP.

m m 25 d’(i’) e’(j’) d 1’ c’(h’) i 1’ e 1’ c 1’

m m 25 d’(i’) e’(j’) d 1’ c’(h’) i 1’ e 1’ c 1’ j 1’ h 1’ B’ X a’(f’) b’(g’) a 1’ f 1’ g 1’ b 1’ Y 45 o e a d b c b 1 a 1 c 1 d 1 50 mm e 1 g 1 j f i g h j 1 f 1 i 1 h 1

PROBLEM NO: 2 Draw the projections of a cylinder 75 mm dia. and 100

PROBLEM NO: 2 Draw the projections of a cylinder 75 mm dia. and 100 mm long, lying on the ground with its axis inclined at 300 to the VP and parallel to the ground.

g’ g 1’(71’) f 1’(61’) e 1’(51’) a 1’(11’) a’ d 1’(41’) b 1’(21’)

g’ g 1’(71’) f 1’(61’) e 1’(51’) a 1’(11’) a’ d 1’(41’) b 1’(21’) c 1’(31’) e’ d’ b’ 1’ 5’ Y 4’ 2’ c’ 3’ h 1 (b 1) g 1 (c 1) f 1 (d 1) e 1 a h (b ) a 1 g( c) f( d) 45 o 6’ 8’ Ф 100 mm X Ф 75 mm f’ h’ e h 1’(81’) 7’ 6 7 1 11 81 (21) 71 (31) 61 (41) 51 8 ) (2 ) (3 ) (4 5

PROBLEM NO: 3 A hexagonal pyramid, base 25 mm side and axis 50 mm

PROBLEM NO: 3 A hexagonal pyramid, base 25 mm side and axis 50 mm long, has an edge of the base on the ground. Its axis inclined at 300 to the ground and parallel to the VP. Draw its projections.

o’ 1 50 mm o’ b’ 1 (a ’ 1 ) c’ 1 (f’

o’ 1 50 mm o’ b’ 1 (a ’ 1 ) c’ 1 (f’ 1) X b‘(a‘) c‘(f‘) d’(e’) 300 e a o 1 d c ’ 1 ) e 1 a 1 o b Y 1 (e f 1 f 25 mm d’ b 1 c 1 d 1

PROBLEM NO: 4 Draw the projections of a cone, base 75 mm dia. and

PROBLEM NO: 4 Draw the projections of a cone, base 75 mm dia. and axis 100 mm long lying on the HP on one of its generators with the axis parallel to the VP.

Ф 100 mm o’ a’ 1 b’ 1(d’ 1) a’ X a Ф 75

Ф 100 mm o’ a’ 1 b’ 1(d’ 1) a’ X a Ф 75 mm b’(d’) d o b c’ c c’ 1 o’ 1 d 1 c 1 a 1 b o 1 Y

INCLINED TO BOTH PLANES PROBLEM NO: 5 A square prism, base 40 mm. side

INCLINED TO BOTH PLANES PROBLEM NO: 5 A square prism, base 40 mm. side and height 65 mm, has its axis inclined at 45˚to the H. P and has an edge of its base on the H. P and inclined at 30˚ to the VP draw its projections.

f’(e’) g’(h’) f’ 1(e’ 1) 65 mm g’ 1(h’ 1) b’(a’) c’(d’) a(e) d(h)

f’(e’) g’(h’) f’ 1(e’ 1) 65 mm g’ 1(h’ 1) b’(a’) c’(d’) a(e) d(h) a 1 c’ 1(d’ 1) d 1 d’ 2 b’ 2 e 1 h 1 g’ 2 f’ 2 a’ 2 40 mm X b’ 1(a’ 1) 450 h’ 2 e’ 2 c’ 2 d 2 c 2 a 2 b(f) c(g) b 1 c 1 f 1 h 2 g 1 g 2 e 2 o 300 f 2 Y

PROBLEM NO: 6 Draw the projections of a hexagonal pyramid edge of the base

PROBLEM NO: 6 Draw the projections of a hexagonal pyramid edge of the base 30 mm and height 65 mm. standing on one of its edges in the HP. The edge making an angle of 45˚ to the VP, and the slant face containing this edge makes an angle of 60˚ to the HP.

o’ o’ 1 a’ 2 ’ 1) 65 mm (a b’ 1 ) (f’

o’ o’ 1 a’ 2 ’ 1) 65 mm (a b’ 1 ) (f’ 1 c’ 1 d’(e’) X f e 35 mm a 600 a 1 f 1 450 f 2 d c b 1 Y 2 2 e 1 d’ 2 e a d 2 o 1 b e’ 2 c’ 2 ’ 1) c‘(f‘) f’ 2 b’ 2 (e d’ 1 b‘(a‘) o’ 2 b 2 c 2 d 1 c 1 o 2

PROBLEM NO: 7 Draw the projections of a cone , base 45 mm. diameter

PROBLEM NO: 7 Draw the projections of a cone , base 45 mm. diameter and axis 50 mm. long , when it is resting on the ground on a point on its base circle, with the axis making an angle of 30˚ to the HP and 45˚ with the VP.

o’ 50 mm o’ 1 o’ 2 Y a 2’ a’ 1 d 2’

o’ 50 mm o’ 1 o’ 2 Y a 2’ a’ 1 d 2’ b 1’(d 1’) c’ 1 Ф 45 mm b c a 1 c 1 o 1 c 2’ c 2 o d 1 β a 2 d 45 o b 2 a c’ ’ b 2 a’ b’(d’) b 1 o 2

THANK YOU

THANK YOU