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

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 ends or bases, parallel to each other and joined by other faces which are rectangles. Faces The imaginary line joining the Centres of the bases or faces is called Axis of Prism. Edge Axis

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 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

(a) Triangular Pyramid: (b) Square Pyramid: (c) Pentagonal Pyramid: 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 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 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 & 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: Apex Edges Generators CONE PYRAMID For Cone and Pyramids, Apex is the point where all the generators or the edges meet.

Rectangle Axis Generators Faces Base Edge CYLINDER PRISM Axis

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

(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: FRUSTUM OF A PYRAMID 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.

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. & P.P. a’,b’ c’,d’ X Y a d o b c

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

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

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

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

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

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

PROBLEM NO: 5 A hexagonal prism has one of its rectangular faces parallel to the ground, its axis is perpendicular to the VP and 3.5cm above the ground. Draw its projections when the nearer end is 2cm in front of VP. Side of the base is 2.5cm, axis 5cm long. 2.5 f (6) e (5) a (1) d (4) b (2) c (3) 3.5 X Y 2 1 2 (6) 5 (3) 4 50 a b (f) c (e) d

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

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

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

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

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

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

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 40mm and axis 65mm long. e’ 40 a’ o’ d’ b’ c’ X Y a b e c d 65 o

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

25mm d’(i’) d1’ i1’ B’ e’(j’) c’(h’) e1’ c1’ j1’ h1’ X g1’ Y a’(f’) b’(g’) f1’ a1’ b1’ 45o c1 e a d b c b1 50mm d1 a1 e1 h1 g1 i1 f1 j f i g h j1

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

Y X 45o a1’(11’) b1’(21’) c1’(31’) d1’(41’) e1’(51’) f1’(61’) g1’(71’) h1’(81’) a’ c’ e’ d’ f’ g’ h’ b’ 1’ 3’ 5’ 4’ 6’ 7’ 8’ 2’ Ф75mm Y X 45o a h (b) g(c) f (d) e 1 8 (2) 7 (3) 6 (4) a1 h1 (b1) g1 (c1) f1 (d1) e1 Ф100mm 5 11 81 (21) 71 (31) 61 (41) 51

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

Y X o’ o’1 50mm b’1(a’1) c’1(f’1) b‘(a‘) c‘(f‘) d’(e’) d’1(e’1) f f1 e 300 d’1(e’1) f f1 e a a1 e1 25mm o o1 d b1 b d1 c c1

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

o’ Ф100mm a’1 b’1(d’1) a’ b’(d’) c’ c’1 o’1 d X d1 Y o c1 a1 o1 Ф75mm a c b b1

INCLINED TO BOTH PLANES PROBLEM NO:5 A square prism, base 40mm INCLINED TO BOTH PLANES PROBLEM NO:5 A square prism, base 40mm. side and height 65mm,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) e’2 h’2 65mm g’1(h’1) f’2 g’2 b’1(a’1) a’2 d’2 c’1(d’1) 450 X b’(a’) c’(d’) b’2 c’2 300 Y d2 a(e) d(h) a1 d1 e1 h1 a2 c2 40mm b2 h2 g1 b(f) c(g) f1 b1 c1 e2 g2 o f2

PROBLEM NO:6 Draw the projections of a hexagonal pyramid edge of the base 30mm and height 65mm. 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.

Y X o’ 65mm o’1 o’2 b’1(a’1) a’2 f’2 c’1(f’1) e’2 b’2 600 d’1(e’1) c’2 450 Y f2 e2 f f1 a2 a1 e1 e d2 a 35mm o1 b2 c2 b b1 d1 d c c1 o2

PROBLEM NO:7 Draw the projections of a cone , base 45mm PROBLEM NO:7 Draw the projections of a cone , base 45mm. diameter and axis 50mm. 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’ 50mm o’1 o’2Y a2’ a’1 d2’ b1’(d1’) b2’ b’(d’) c’ c’1 a’ c2’ β d 45o d1 a2 d2 Ф45mm o a c a1 c1 c2 o1 b2 b b1 o2

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