Sections of Solids aS 13.1 INTRODUCTION In the engineering practice, its often required to make the drawing showing the interior deta, the object. Ifthe object is simple in its construction, the interior portion of the object can be pe interpreted by dotted lines in the orthographic projections. When the dotted lines of hidden partsay, too many, the views become more confusing and hard to read. In such cases, views can be drawnty cutting the object by an imaginary cutting plane so as to expose its interior or hidden details. The part of the object between the cutting plane and the observer is assumed to be removed so as to show the internal constructional features or details of the invisible surface. The exposed interior detail ae drawn in continuous thin lines instead of dotted lines. Such views are known as sectional view views in section. The section surfaces are indicated by section lines, evenly spaced and inclined at 5° to the reference line. 13.2 SECTION PLANES ‘These are generally perpendicular planes. These may be perpendicular or parallel to one of he principal planes and either perpendicular, parallel or inclined to the other plane. These planes ar usually described by their traces. 13.3 SECTIONS Basically, sections are of two types : ( Apparent Section (i) True Section (Apparent Section. The projection of the section on the principal plane to which the section plane is perpendicular, a straight line coinciding with the trace of the section plane onit. Its projection on the other plane to which it is inclined is called apparent section. (ii) True Section. The projection of the section on a plane parallel to the section plane shows the true shape of the section. When the section plane is parallel to the HP or ground plane, the true shape of the section is seen in the sectional top view. When it is parallel to the VP, the true shape is proj in the sectional front view. But when the section plane is inclined to one of the principal planes, the section has tobe projected onan auxiliary plane parallel to the section plane, to obtain its true shape. When the section plane is perpendicular to both the principal planes, the sectional side view shows the true shape of the section. 43.4 FRUSTUM OF A SOLID AND A TRUNCATED SOLID When a cone or a pyramid is cut by a plane parallel to its base, thus removing the top portion, remaining lower portion is called its frustum of a solid as shown in the Fig. 13.1. The section obi is called the true section of a solid. the Scanned with CamScannerSECTIONS OF SOLIDS 281 Cutting plane parallel to the base Fig. 13.1 Frustum of a solid When a solid is cut by a plane inclined to its base, thus removing the top portion, the remaining lower portion is called its truncated of a solid as shown in the Fig. 13.2. The section obtained is called the apparent section of a solid. Cutting plane Inclined to the base Fig. 13.2 Truncated of a solid 13.5 CLASSIFICATION OF SECTIONS OF SOLIDS The solids may be assumed to be cut by the section planes in many ways to obtain the sectional views. These are as follows : * Section plane parallel to the HP * Section plane parallel to the VP * Section plane perpendicular to the VP and inclined to the HP * Section plane perpendicular to the HP and inclined to the VP * Section plane perpendicular to both HP and VP 13.6 SECTION PLANE PARALLEL TO THE HP sothe HP therfore ts VI willbe a As the secti is perpendicular to the VP and parallel to the HP, therefore its VT straight tne paral aon fT ‘As the section plane is parallel to the HP, projection ee ie section on the HP is trae shape and siz. Its projection on the VP isalline and coincides wi ° Plane. Scanned with CamScanner282 ENGINEERING DRAWING AND COMPUTER GRAPHiog PROBLEM 13.1 A right regular pentagonal pyramid, edge of base 30 mm and height 50 mm, rests ong base on HP with one of its base edges perpendicular to VP. A section plane parallel to the HP cuts the Pyramig bisecting its axis. Draw its front view and sectional top view. (PTU, Jalandhar December 2003) SOLUTION. () Draw the projections of the pentagonal pyramid in the required position and label it. (i) As the section plane is parallel to the HP and perpendicular to the VP. Hence itis representa by its VT. ; (i) The slant edges 0'1’,02','3',0'4' and ('5' intersect at the points a,b,c’, d” and ¢ respectively in the front view. (iv) Project these points on the corresponding edges in the top view. Join these points in p order and draw section lines int. It will give the required sectional top view as shown in Fi 13.3. iB Fig. 13.3 Solution to problem 13.1 PROBLEM 13.2 A right regular hexagonal pyramid, edge of base 30 mm and height 50 mm, rests on its base on ground plane with one of its base edges parallel to VP. A section plane parallel to the HP cuts the pyramid bisecting its axis Draw its front view and sectional top view. SOLUTION. () Draw two lines xy ang g/ by a suitable distance apart. Draw the projections of the hexagonal Pyramid in the required position and label it. Scanned with CamScannerSECTIONS OF SOLIDS . 283 (i) Draw the VT to represent the section plane at a distance of 25 mm from gl and parallel to it 4 3.5" 2.6 v Fig. 13.4 Solution to problem 13.2 . vey int (iii) The slant edges 0'1’, 0'2', 0’3', 0'4', 0'5’, 0'6 cut the points a,U,,c',d’,e’,f’ respectively in the front view. : ; iv) Project i it i i in the top view. Join these (io) these points of intersection on the corresponding edges in the , : points in ae ‘order and draw section lines init. It will give the required sectional top view as shown in Fig. 13.4. / i id, si nd height 65 mm, lies on one _ PROBLEM 13.3 A right regular pentagonal pyramid, side of base 30 mm a 1 ofits triangular face in HP such tha ite axis parallel to VP. A section plane parallel to HE, cuts the axis ata Point 10 mm away from its base. Draw its front view and sectional top view. (PTU, Jalandhar May 2004) SOLUTION. ( Draw the projections of the given pyramid in the required position and label it. Scanned with CamScanner284 : ENGINEERING DRAWING AND COMPUTER GRAPHicg (ii)_As the section plane is parallel to HP and perpendicular to the VP. Hence it is. tepresented by its VT. ; | (iii) In the front view, at a distance of 10 mm from its base, find the point along its axis and draw a cutting plane line coinciding with VT, passing through the above mentioned point, (iv) Mark the points of intersection of the section plane with different edges of the Pyramid. (2) Project these points on the corresponding edges in the top view. Join these points in their proper order and draw section lines init. It will give the required sectional top view as shown in Fig. 13.5. Fig, 13.5 Solution to problem 13.3 PROBLEM 13.4 A right circular cone, diameter of elements in HP, such that its axis is parallel to VP. A sect VP cuts the cone, meeting the axis at a distance of 15 mm view. SOLUTION (i Draw the projections of the given cone in the (ii) Asdiscussed above, in the front view, base 50 mm and height 60 mm, lies on one of its ion plane parallel to the HP and perpendicular to the from the base. Draw its front view and sectional top required position and label it. ata distance of 15 mm from its base, find the point along its axis and draw a cutting plane line coinciding with VT, Passing through the given point. (tii) Mark the points of intersection of the section plane with different elements ‘or base of the cone. (iv) Project these points on the corresponding elements or base in the top view. Join these points in their proper order and draw section lines in it. It will Bive the required sectional top view as shown in Fig. 13.6. Scanned with CamScannerTIONS OF SOLIDS SECTIOI 285 eL pM eae] Bs] ay 44 Fig. 13.6 Solution to problem 13.4 PROBLEM 135 A right regular hexagonal prism, side of base 20 mm and length of axis 55 mm, is lying on one of its rectangular faces in HP. Its axis is parallel to both HP and VP. It is cut by a section plane parallel! toand at a distance of 20 mm from the HP. Draw its front view and sectional top view. SOLUTION. () Draw the projections of the given prism in the required position and label it. Here in this case, the projections will be started from the side view and then project the frontand top views from. it (i) Draw a cutting plane coinciding with VT parallel to xy and 20 mm away from it. (ii) Mark the points of intersection of the section plane with different ‘elements of the prism. (io) Project these points on the corresponding elements in the top view. Join these points in their Proper order and draw section lines in it. It will give the required front and sectional top views as shown in Fig. 13.7. PROBLEM 13.6 A triangular prism, side of base 45 mm and length of axis 60 mm, is lying on one ofits rectangular faces in HP. Its axis is parallel to both HP and VP. It is cut bya section plane parallel to and at a distance of 20 mm from the HP. Draw its front view and sectional top view. (PTU, Jalandhar December 2006, December 2007) SOLUTION. The procedure followed to solve this problem is same as explained in problem 13.5. interpretation of the solution is left to the student. See Fig. 13.8. Scanned with CamScanner286 ENGINEERING DRAWING AND COMPUTER GRAPHICg s y Fig. 13.7 Solution to problem 13.5 Fig. 138 Solution to problem 13,6 Scanned with CamScannerSECTIONS OF SOLIDS 287 PROBLEM 13.7 A right circular cylinder dianteter of base 50 mm and height 60 mm, rests om its base rim on ground plane such that its axis is inclined at 45°to the ground plane and is parallel to the VP. Assection plane parallel to the HP cuts the cylindér bisecting its axis, Draw its front view and sectional top view. ; (PTU, Jalandhar December 2014) SOLUTION. (Draw two lines xy and gi by a suitable distance apart. Draw the projections of the cylinder in the required position and label it. (ii) Draw the VT to represent the section plane at a distance of 30 mm from base and parallel to it. (iii) The section plane line cuts the elements 1,'a,’, 2,'b,’, so on, at points I’, H’ and so on. Project these points on the corresponding elements in the top view. Join these points in proper order and draw section lines in it. It will give the required sectional top view as shown in Fig. 13.9. 50 1213 14 2m Mim, q 15. 7 B IR 16 LN ' : S of fo. a D a, a] \S y 2 ® P, 5 7 A 3 ANS, * e ee LP. y ir ks el] | ty] em] ain spb | a” g 62 41a 1 Fig. 13.9 Solution to problem 13.7 Scanned with CamScanner288 ENGINEERING DRAWING AND COMPUTER GRAPHICS PROBLEM 13.8 A hexagonal prism of base edge 20 mm long and height 60 mm is resting on one of its corners on HP with the base making 60° to HP. Axis is parallel to the VP. A section plane parallel to HP and perpendicular to VP cuts the object such that it is 15 mm away from the base measured along its axis. Draw the front view and sectional top view of the solid. (PTU, Jalandhar December 2002) SOLUTION. (Draw the projections of the prism in the given position and label it. (i) Draw the section plane line VT parallel to the xy and 15 mm away from the base along its axis, (iii) Mark the points of intersection of the section plane with different elements of the prism. (2) Project these points of intersection say ¢,, $1, iy, hy, ¢, in the top view to their corresponding elements of the prism. Join these points in their proper order and draw section lines init. It will give the required sectional top view as shown in Fig. 13.10. Fig. 13.10 Solution to problem 13.8 Scanned with CamScannerSECTIONS OF SOLIDS PROBLEM 13.9 A right regular hexagonal prism, side of base 20 mm and height 45 corner : 7 Ly mm rests ona ofits base on the HP ‘with the longer: ‘edge passing through this corner: ‘making an angie of 30° to the HP. A section plane parallel to the HP cuts the axis at a distance of 15 mm from top base. Draw its sectional top view and front view. 289 SOLUTION. The interpretation of the solution is left to the reader. See Fig. 13.11. d'_cie’ obit a’ vs | _t i ¢ i i ja isso” 2.6 |r E 2 |_|r 56. ler Ie | | /s ? 7 j la a a Fig. 13.11 Solution to problem 13.9 PROBLEM 13.10 A right regular hexagonal pyramid, edge of base 30 mm and height 60 mm, rests on its base in HP with one of its base edges perpendicular to VP. A section plane parallel to the HP cuts the pyramid bisecting its axis. Draw its front view and sectional top view. SOLUTION. () Draw the projections of the pyramid in the given position and name the comer Points on it. (i) Draw the cutting plane line VT parallel to xy and bisecting its axis. (ii) The cutting plane live VT cuts the various edges as shown in Fig. 13.12. (iv) Project these points in the top view. The projections c, fof c’f in the top view are to lie on 03. and 06. By direct intersection these points cannot be plotted. Project c’f’ horizontally on a slant edge (which gives true length) or 0'1’,(0’2’,) and then project point of intersection vertically intoin the top view. With 0 as centre, rotate these points to lie on 03 and 06 at cand frespectively. () Join the points in proper order and draw section lines in it. Scanned with CamScanner\ / 2s ENGINEERING DRAWING AND COMPUTER GRAPHicg s Fig, 13.12 Solution to problem 13.10 PROBLEM 13.11 A right regular pentagonal prism, side of base 25 mm and length of axis 60 mm, is lying on one of its rectangular faces in HP. Its axis is parallel to both HP and VP. It is cut by a section plane parallel to and at a distance of 15 mm from the HP. Draw its front view and sectional top view. SOLUTION. The interpretation of the solution is left to the reader. See Fig. 13.13. s i & | L I t y Fig. 13.13 Solution to problem 13.11 Scanned with CamScannerSECTIONS OF SOLIDS = 291 PROBLEM 13.12 A right regular pentagonal pyramid, ed: P ; : , edge of base 30 mm and height 60 mm, rests on its base on HP with one of its base edges parallel to VP. A section plane parallel to the HP cuts the pyramid bisecting its axis. Draw its front view and sectional top view. (PTU, Jalandhar May 2009) SOLUTION. All the construction lines are retained to make the solution self-explanatory. See Fig. 13.14. 7 . Fig. 13.14 Solution to problem 13.12 13.7 SECTION PLANE PARALLEL TO THE VP ‘As the section plane is perpendicular to the HP and parallel to the VP, therefore its HT will be a straight line parallel to xy and has no VT. As the section plane is parallel to the VP, projection of the ie the VP is true shape and size. Its projection on the HP isa line, coinciding with the HT of pl a PROBLEM 13.13 A right regular pentagonal pyramid, edge of base 30 mm and height 55 mm, rests on its base on HP, such that one ofits base edges is perpendicular to the VP. A section plane parallel to the VP cuts the pyramid at a distance of 10 mm from the axis. Draw its top view and sectional front view. (PTU, Jalandhar June 2003, May 2004) SOLUTION. (Draw the projections of the given pyramid in the required position and label it. (ii) As the section plane is parallel to VP and perpendicular to the HP, hence it is represented by its HT. In the top view, cut a distance of 10 mm from its axis, coinciding with HT. (iii) Mark the points of intersection of the section plane with different edges of the pyramid. (i) Project these points on the corresponding edges in the front view. Join these points in proper order and draw section lines init. Iewill give the required sectional front view as shown in ig. 13.15. Scanned with CamScanner292 ENGINEERING DRAWING AND COMPUTER GRAPHicg SS SSS NS SS WSS SA a Fig. 13.15 Solution to problem 13.13 PROBLEM 13.14 A right regular pentagonal pyramid, edge of base 30 mm and height 50 mm, rests onits base on ground plane, such that one ofits base edges perpendicular to VP. A section plane parallel to the VP cls the pyramid at a distance of 10 mm from the axis, Draw its top view and sectional front view. SOLUTION. The interpretation of the solution is left to the student. See Fig. 13.16. PROBLEM 13.15 A right pentagonal pyramid of base side 25 mm and height 50 mm rests on the HP with one edge of the base at 45° to the VP. Draw the sectional elevation ofthe solid is cut by a plane parallel to the VP containing the apex. (PTU, Jalandhar December 2004) SOLUTION. (Draw the projections of the given pyramid in the required position and label it. (i) As the section plane is parallel to the VP, hence it is represented by its HT. In the top view, draw a cutting plane line passing through the apex coinciding with the HT. (iii) Mark the points of intersection of the section plane with different edges of the pyramid. (iv) Project these points on the corresponding front view. Join these points in the proper sequent? and draw section lines in it as shown in Fig. 13.17. Scanned with CamScannerSECTIONS OF SOLIDS 293 Fig. 13.16 Solution to problem 13.14 Fig. 13.17 Solution to problem 13.15 PROBLEM 13.16 A cube of 35 mm long edge is resting on the HP on one of its faces with a vertical face inclined at 30° to the VP. It is cut by a section plane parallel to the VP and 10 mm away from the axis and further ‘away from the VP. Draw its sectional front view and top view. (PTU, Jalandhar December 2007, May, 2008, May 2013) SOLUTION. () Draw the projections of the cube in the required Position and label it. (ii) As the section plane is parallel to the VP and is perpendicular to the HP, hence the section plane is represented by its HT. Draw a line HT in the top view parallel to.xy and 10 mm from its axis, (iii) Mark the points of intersection of the section plane with different edges of the cube. (2) Project these points on the corresponding edges in the front view and join these points in the correct sequence and draw section lines in it as shown in Fig. 13.18. Scanned with CamScanner294 ENGINEERING DRAWING AND COMPUTER GRAPHiog Fig. 13.18 Solution to problem 13.16 PROBLEM 13.17 A right regular pentagonal pyramid, edge of base 30 mm and height 60 mm, rests HP on one ofits comers. Its base is inclined at 45° to HP and axis is parallel to the VP. A section plane paral to the VP cuts the pyramid at a distance of 10 mm from its axis. Draw its top view and sectional front view, SOLUTION, () Draw the projections of the given pyramid in the required position and label the comer points. (Draw the section plane line HT, to represent the section plane in top view, coinciding withHT and passing through a distance of 10 mm from its axis as shown in Fig. 13.19. (iti) Project these points of intersection on the corresponding front view. Join these points in Proper order and draw section lines in it. Complete the required top view and sectional front view. PROBLEM 13.18 A right circular cylinder diameter of base 40 mm and height 60 mm, is lying on HPon one of its elements, such that its axis is inclined at 30° to the VP. A section plane parallel to VP, cuts the cylinder ‘ata distance of 10 mm from its end face meeting its axis. Draw its sectional front view and top view. SOLUTION. (9 Draw the projections of the given cylinder in the required position and label it. (i) Draw the section plane line HTT, to represent the section plane in top view, at a distance of 10 mm from its end face and parallel to the xy. (iti) Mark the points of intersection of the section plane with different elements of the cylinder. (iv) Project these points on the corresponding elements in the front view. Join these points in proper order and draw section lines in it. The required sectional front view and top view is shown in Fig. 13.20. Scanned with CamScannerSECTIONS OF SOLIDS 295 ° ’ G : 4 i ¢ , | 8 ‘ | | i ! . : ‘ va 4 8 ‘ 3 : “0 , : 42 19a wat bd pi, Be 7 = oT. , met | N° 10, 6 10'-| a 0 is," 9 ot | Z Hh 8 fy i? y . Gi a i y 30° side | sas | asally 8 108 aT Fig. 13.20 Solution to problem 13.18 Scanned with CamScanner296 ENGINEERING DRAWING AND COMPUTER GRAPHIcg PROBLEM 13.19 A right circular cone diameter of base 50 mm oe ae o ay fet a = o is elements in HP with its axis parallel to the VP. A section plane perpendicu ae eee > cuts the cone and is 10 mm away from the axis, Draw its top view and secti poker SOLUTION. The interpretation of the solution is left to the students. See Fig. 13.21. o Y vB: 2, 16) 3518) 4,14" 8 5,13 oy Fae s : 1 Ty 4 x TH tN 8 1 A 0, ' e| 2 T 3 7 4, ae) 5 fcr Fig. 13.21 Solution to problem 13.19. PROBLEM 13.20 A right regu ar hexagonal pyramid, edge of base 25 mm artd height 60 mm rests on the HP with one edge ofthe base at 45°to the VP. A section plane parallel to the VP, cuts the pyramid at a distance of 12 mm from the axis. Draw its top view and sectional front view. SOLUTION. (© Draw the projections of the given pyramid in the Tequired position and label it, (In the top view, cuta distance of 12 mum from its axis, coinciding with HT. Scanned with CamScannerSECTIONS OF SOLIDS : 297 3 AN ~ “NS SV SS GE S Fig. 13.22 Solution to problem 13.20 PROBLEM 13.21 A right regular hexagonal prism, side of base 25 mm and height 60 mm, lies on one of its rectangular faces on HP and its axis inclined at 45° to the VP. A section plane parallel to the VP cuts the prism bisecting its axis, Draw its top view and sectional front view. SOLUTION. ‘All the construction lines are retained to make the solution self-explanatory. See Fig. 22, 13.8 SECTION PLANE PERPENDICULAR TO THE VP AND INCLINED TO THE HP When a section plane passing through a solid is perpendicular to the VP and inclined to the HP, its VTi inclined to the xy and HT which is perpendicular to the xy serves no purpose in drawing the section views, so it is omitted. The projection of such a section in front view is a line, coincident with the cutting plane line VT. ‘As the section plane is inclined to HP its projection on the HP does not show its true shape and size and is called apparent section. ~The true shape of section may be obtained on an auxiliary inclined plane (ATP) parallel to the given section plane. Scanned with CamScanner288 ENGINEERING DRAWING AND COMPUTER GRAPHicg << = as. a L <2 = i S as ZF is Fig. 13.23 Solution fo problem 13.21 Scanned with CamScannerSECTIONS OF SOLIDS 299 PROBLEM 13.22 A cylinder of 45 mm diameter and 60 mm long is resting on one of its bases on HP. It is cut by a section plane inclined at 60° with HP and perpendicular to VP passing through a point on the axis 15 1mm from it top end. Draw its front view, sectional top view and true shape of the section. a i (PTU, Jalandhar December 2009) SOLUTION. The interpretation of the solution is left to the reader. See Fig. 13.24. > Fig. 13.24 Solution to problem 13.22 PROBLEM 13.23 A right circular cone, diameter of base 50 mm, height 60 mm, rests on its base on HP. A section plane perpendicular to VP and inclined to HP at 30°, cuts the cone bisecting its axis, Draw its front view, Sectional top view and true shape of the section. (PTU, Jalandhar May 2006) SOLUTION. () Draw the projections of the cone in the given position and name the points on it. (i) Draw the cutting plane line VT inclined at 30° to xy and bisecting its axis. (iii) The cutting plane line VT cuts the various elements as shown in Fig. 13.25. (#2) Project all the points on the corresponding elements in the top view. Join these points by a smooth curve and draw section lines in it. (2) To draw the true: shape of the section, draw a new reference line x,y, at a convenient distance and parallel to the cutting plane line. Scanned with CamScanner= ENGINEERING DRAWING AND COMPUTER GRAPHicg i ion i it i jectors tox,y,. On the (vi) Through the points on the section in front view, draw dicular proje Yt perpendicula projector through 1’, from x,y, cut the distance of point 1 in the top view from xy. Fig. 13.25 Solution to problem 13.23 Similarly plot the other points. Join these points by a smooth curve and draw section lines in it. The required figure is called the true shape of the section. PROBLEM 13.24 A right regular hexagonal pyramid, edge of base 25 mm, height 50 mm, rests on its base on HP, with one ofits base edges parallel to VP. A section plane perpendicular to VP and inclined to HP at 30°, cuts the Pyramid, bisecting its axis. Draw its front view, sectional top view and true shape of the section. SOLUTION. (9 Draw the projections of the pyramid in the given position and name all the comer points on it (i) Draw the cutting plane line VT inclined at 30° the xy and bisecting its axis, (ii) The cutting plane line VT cuts the various edges as shown in Fig. 13.26. Scanned with CamScannerSECTIONS OF SOLIDS 301 Fig. 13.26 Solution to problem 13.24 (io) Project all these points on the corresponding edges in the top view. Join these points in proper order and draw section lines in it. (©) Todraw the true shape and size of the section, draw anew reference line x,y,, ata convenient Position and parallel to the cutting plane line. (©) Through the points on the section in front view, draw perpendicular projectors ox,, On the Perpendicular projectors cut the distances of the points from xy in the top view. Join these a in proper order and draw section lines in it. This is the required true shape of the lon, Scanned with CamScanner302 ENGINEERING DRAWING AND COMPUTER GRAPHicg. PROBLEM 13.25 A square pyramid, edge of base 35 mm, height 50 mm rests on its base on HP with its base edges equally inclined to VP. A section plane perpendicular to the VP and inclined to the HP at 30°, cuts the pyramid bisecting its axis. Draw its front view, sectional top view and true shape of the section. (PTU, Jalandhar December 2013) SOLUTION. () Draw the projections of the pyramid in the given position and name the comers points on it, (i) Draw the cutting plane line VT inclined at 30° to xy and bisecting its axis. (iii) The cutting plane line VT cuts the various edges of the pyramid as shown in Fig. 13.27, Fig. 13.27 Solution to problem 13.25 Scanned with CamScannerSECTIONS OF SOLIDS 303 (iv) Project these Points in the top view. The Projections b, d of b’,d’ in the top view are to lie on 02 and 04. By direct intersection these points cannot be plotted. Project b’, d’ horizontally on a slant edge (which gives true length) 0'1’ or 0°3' and then Project the point of intersection vertically into 01 or 03 in the top view. With as centre, rotate these points to lie on 02 and 04 at b and d respectively. (©) Join these points in proper order and draw section lines in it. (vi) To draw the true shape of the section, draw anew Teference line x,y, at a convenient position and parallel to the cutting plane line. (vii) hie e et tae ee oo in the earlier problems, Repeat the same procedure here to 8 ie shape of the section. PROBLEM 13.26 A square pyramid, edge of base 30 mm, height 45 mm rests on its base on HP with its base edges equally inclined to the VP. A section plane perpendicular to the VP and inclined to the HP at 30°, cuts the pyramid bisecting its axis. Draw its front view, sectional top view, sectional left side view and true shape of the section. (PTU, Jalandhar May 2006, December 2015) SOLUTION. The interpretation of the solution is left to the student. See Fig. 13.28. o o" fa Fig. 13.28 Solution to problem 13.26 Scanned with CamScanner304 ENGINEERING DRAWING AND COMPUTER GRAPHicg PROBLEM 13.27 A right regular pentagonal pyramid, edge of base 25 mm and height 50 mm, rests on its base on HP such that one ofits base edges to be perpendicular to VP. A section plane perpendicular to the VP ang inclined to the HP at 30° cuts the pyramid bisecting its axis. Draw its front view, sectional top view and true shape of the section. (PTU, Jalandhar May 2019) SOLUTION. (Draw the projections of the pyramid in the given position and label it. (ii) Draw the cutting plane line VT inclined at 30° to xy and bisecting its axis. (ii) The cutting plane line VT cuts the various edges of the pyramid as shown in Fig. 13.29. (iv) Projectalll the points of intersection in the top view. Join these point in proper order and draw section lines in it. (v) To draw the true shape of the section, draw a new reference line x,y, ata convenient distance and parallel to the section plane line. (vi) Project all the points as discussed in earlier problems. Repeat the same procedure to get the true shape of the section. Fig, 13.29 Solution to problem 13.27 PROBLEM 13.28 A cylinder of 45 mm diameter and 60 mm long is resting on one of its bases on HP. It is cut by a section plane inclined at 60° with HP and perpendicular to VP passing through a point on the axis 15 mum from its top end. Draw its sectional top view, front view and sectional end view. (PTU, Jalandhar May 2001) SOLUTION. (i Draw the projections of the cylinder in the given position and label it. Scanned with CamScannergECTIONS OF SOLIDS 05 ii) Draw the cutting plane line VT inclined at 60° : . ' (0) 5mm from the top end of thecylinden 60° to xy and passing through a point on the axis (i) The cutting plane line VT cuts the various edg, (jo) Project the points of intersection in the top Join these points in proper order and dr: es as shown in Fig. 13.30. view and end view to their corresponding edges. ‘aw section lines in it. « ’ el] 7 bt Jer 7, 3 | ¥; + e id',h, / a My ely ip ; - 4 AL a : srqe | Se | ayer SArasy 10S aes” PAD Vv 14 5 i 46 of 1 bel 2 7 3 sobs. 45 y Fig. 13.30 Solution to problem 13.28 PROBLEM 13.29 A cube of 65 mm long edges has its vertical faces equally inclined to the VP. It is cut by section plane, perpendicular to the VP, so that the true shape of the section is a regular hexagon. Determine the inclination of the cutting plane with the HP and draw the sectional top view and true shape of the section. (PTU, Jalandhar June 2003) SOLUTION. (@ The true shape of the section is a regular hexagon, as all the edges of the cube are equal, therefore the section plane should pass through the mid-points of all the six edges, which are cut by a section plane. (i) The cube must be kept in such a way that its vertical faces make equal angles with the VP. Therefore draw the top and front views of the cube in this position. Scanned with CamScanner306 ENGINEERING DRAWING AND COMPUTER GRAPHiog (ii) The cutting plane now passes through the points f, 8, eand hwhich are the mid-points of edges in the top view. Project these points in the front view to get the section pI jane through fy and é’h’. , ; (jo) Join the points ,b, fg, j,hin the top view by straight lines to obtain the sectional top view ang draw the section lines in it. ine at any conventent Sane Then draw x,y; line parallel to the section plane line at any conVe™l At the true shape of the coe i be drawn on an auxiliary top view as shown in Fig. 13.31. y, (4: Fig. 13.31 Solution to problem 13.29 PROBLEM 13.30 A right regular hexagonal pyramid, base 30 mm side and axis 65 mm long, srs! nits base on the HP with one ofits base edges parallel othe VP. Itis cut by section plane, perpendicular VP and inclined to the HP at 60° and intersecting the axis at a point 25 mm above the base. Draw the front vi sectional top view and sectional left side view. SOLUTION. The interpretation of the solution is left to the reader. See Fig. 13.32. Scanned with CamScannerSECTIONS OF SOLIDS 307 “a Lis _ 7 WOK eS A Z gy] age y Fig. 13.31 Solution to problem 13.29 PROBLEM 13.30 A sqiuare pyramid, edge of base 40 mm and height 60 mm, is resting on its base in HP with one of its base edges perpendicular to VP. A section plane perpendicular to the VP and inclined to the HP cuts the pyramid in such a way that the true shape of the section is a trapezium where parallel sides measure 30 mm and 15 mm. Draw the front view, sectional top view and true shape of the section. Also determine the inclination of the section plane with HP. SOLUTION. The interpretation of the solution is left to the student. See Fig. 13.32. PROBLEM 13.31 A square prism edge of base 25 mm and 70 mm long axis is resting on its base in HP. The edges of the base are equally inclined to the VP. It is cut by a plane inclined to the HP and perpendicular to the VP passing through-the mid-point of the axis in such a way that the true shape of the section is rhombus having diagonals of 70 mri and 35 mm. Draw the projections and determine the inclination with HP. SOLUTION. () Draw the projections of the square prism in the given position and name it. (i) Mark the mid-point of the axis in the front view. With the mid-point of the axis as centre and radius equal to 35 mm, draw an arc cutting the two opposite sides of the prism as shown in Fig. 13.33, (iil) Project these points in the top view and complete the true shape of the section. Draw the Section lines in it. Scanned with CamScannerHh oN | Mr. Gr ly q ld x AG 5 A -———_ : Fig. 13.33 Solution to problem 13.31 Scanne d with CamScannerSECTIONS OF SOLIDS se PROBLEM 13.33 A right circular cylinder diameter ‘of bas is i s . nate se 60 mm and. in HP. It is cut by a section plane, which is perpendicular to the VP and EI SHERI ape Te slnder bisecting its axis. Draw the front view, sectional top view, sectional left side view and true pay of the section. PTU, SOLUTION. The interpretation of the solution is left to the a cane popkmane om” g 7,11 5,13" 4,14" 1 13 to,16 9,1" 8,2 79g 4 Pa By) 7 F | > 4) ral . L\\ tr} , ef 7 Th" 30°) x ° = KJ 14 1 EN aL! 15 110) f y, 9 . 2 a ND Pp 60 3 a 6 + . a E > x y c, 1, 4, h @, gt Fig. 13.35 Solution to problem 13.33 Scanned with CamScannerENGINEERING DRAWING AND COMPUTER GRAPHicg oe oe PROBLEM 13.34 A right regular hexagonal pyramid, edge of base 20 ma fe begin mm, ie ts ni its base in HP, with one ofits base edges perpendicular to VP. A section plane incline cee an repens i VP cuts the pyramid, bisecting its axis. Draw its front view, sectional top view Tai 310 SOLUTION. : ; (Draw the projections of the pyramid in the given position and name the corner points on it, (i) Draw the cutting plane line VT inclined at 30° to xy and are Pe ee (iif) The cutting plane line VT cuts the various edges of the pyramid a: ‘ A Fig. 13.36 Solution to problem 13.34 (2) Project these points in the top view. The projections c, fof ¢,f are to lie on 03 and 06. By direct, intersection, these points cannot be plotted. Project ¢’, f horizontally on a slant edge (which gives true length) 0’1,’(0’2’,) and then project point of intersection vertically in the top view. With 0 as centre, rotate there points to lie on 03 and 06 at cand f respectively. (2) Join these points in proper order and draw section lines in it. (@i) To draw the true shape of the section, draw a new reference x, y, ata convenient position and parallel to the cutting plane line. (ii) Project all these points as discussed in the previous problems. Repeat the same procedure t0 get the true shape of the section. Scanned with CamScannerSECTIONS OF SOLIDS 311 PROBLEM 13.35 A square i ofits base edge inclined at 30° to the VP. ee of bse 30 mm, height 60 mm ress on its base on HP with one dats the pyramid ata distance of 25 mm from its oe plane perpendicular to the VP and inclined to the HP at 45° shape of the section. ‘apex on the axis. Draw its front view, sectional top view and true SOLUTION. All the construction li a7 iction lines are retained to make: the solution self-explanatory. See Fig. 4,5" 316" 2.1" my, 3 Cy 1 3 Fig. 13.38 Solution to problem 13.36 Scanned with CamScannerod ENGINEERING DRAWING AND COMPUTER GRAPHICS i i is 60 mm long is resting on HP PROBLEM 13.36 A right regular hexagonal prism of base 30 mm and axis mn rest on one of its base edge with its axis inclined at 45° to HP and parallel to the VP. A section ere HP at 60° and perpendicular to the VP is passing through the topmost edges of the prism. view, sectional top view and true shape of the section. / : SOLUTION. All the construction lines are retained to make the solution self-explanatory. See Fig, 13.38. ; _ , PROBLEM 13.37 A cube of 55 mm edge rests on HP with one ofits vertical faces inclined at 30° to the VP, section plane inclined to the HP at 60° and perpendicular to the VP cuts the solid at a distance of 48 mm from the base along its axis. Draw its front view, sectional top view and true shape of the section. ; SOLUTION. All the construction lines are retained to make the solution self-explanatory. See Fig, 13.39. \Y Fig. 13.39 Solution to problem 13.37 13.9 SECTION PLANE PERPENDICULAR TO THE HP AND INCLINED TO THE VP When a section plane passing through a solid is perpendicular to the HP and inclined to the VP, its HT is inclined to the xy and VT which is perpendicular to the xy serves no purpose in drawing the section views, so itis omitted. The projection of such a section in top view isa line, coincident with the cutting plane line HT. As the section plane is inclined to VP, its projections on the VP does not show its true shape and size and is called apparent section. The true shape of the section may be obtained on an auxiliary vertical plane (AVP) parallel to the given plane. PROBLEM 13.38. A right regular hexagonal pyramid, edge of base 25 mm, height 50 mm lies on one of its slant edges on HP with its axis parallel to VP. A section plane perpendicular to HP and inclined to VP at 30° cuts the pyramid bisecting its axis. Draw its top view, sectional front view and true shape of the section. SOLUTION. All the construction lines are retained to make the solution self-explanatory. See Fig 13.40. Scanned with CamScannerSECTIONS OF SOLIDS an Fig. 13.40 Solution to problem 13.38 PROBLEM 13.39 A right regular hexagonal pyramid, edge of base 25 mm, height 60 mm, rests on its base on HP with one of its base edges parallel to VP. A section plane perpendicular to HP and inclined to the VP at 30° cuts the pyramid and is 10 mm away from the axis, Draw its top view, sectional front view and true shape of the section. SOLUTION. () Draw the projections of the pyramid in the given position and name the corner points on it. (Draw the cutting plane line HT inclined at 30° to xy and 10 mm away from the axis. This can be done by drawing a circular arc of 10 mm radius with oas centre and draw the cutting plane line HT tangential to the arc. (iii) The pointsa,b, c,d, of the cutting plane line intersect with various edges of the pyramid in the top view. (iv) Project these points in the front view to their corresponding edges. Join these points in proper order and draw section lines in it. As the section plane is inclined to the VP, projections of the section in the front view is an apparent section. To draw true shape of the section, draw an auxiliary plane parallel to the section plane, as, shown in the Fig. 13.41 by the method already explained. ©) Scanned with CamScanner314 : ENGINEERING DRAWING AND COMPUTER GRAPHicg Wy WK OK Fig. 13.41 Solution to problem 13.39 PROBLEM 13.40 A right regular pentagonal pyramid, edge of base 25 mm and height 50 mm, rests on its base on HP such that one of its base edges to be perpendicular to VP. A section plane perpendicular to the HP and inclined to the VP at 30° cuts the pyramid and is 10 mm away from the axis, Draw its top view, sectional front view and true shape of the section. (PTU, Jalandhar December 2008) SOLUTION. () Draw the projections of the pyramid in the given position and label it. (i) Draw the cutting plane line HT inclined at 30° to xy and 10 mm away from the axis. This can be done by drawing circular arc of 10 mm radius with 0 as centre and draw the cutting plane line HT tangential to it. Scanned with CamScannerSECTIONS OF SOLIDS : 315 (if) The points of intersection a, b,c, d of the cutting plane line with various elements of the pytamid in the top view are sho wn in Fig. 13.42, Fig. 13.42 Solution to problem 13.40 (i) Project these points in the front view to their corresponding elements. Join these points in proper order and draw section lines in it. (0) To get the true shape of the section, draw an auxiliary plane parallel to the section plane, Project all the points as discussed in the previous problems. PROBLEM 13.41 A right circular cone, diameter of base 50 mm and height 65 mm, rests on its base in HP. A section plane perpendicular to the HP and inclined to the VP at 45° cuts the cone and is 12 mm away from the axis. Draw its top view, sectional front view and true shape of the section. (PTU, Jalandhar December 2004) SOLUTION. () Draw the projections of the cone in the given position and label it. (i) Draw the section plane line HT inclined at 45° to xy and 12 mm away from the axis. This can bedoneby drawing a circular arc of 12 mm radius with o as centre and draw the section plane line HT tangential to it. Scanned with CamScannera ENGINEERING DRAWING AND COMPUTER GRAPHIcs, ie b : = v 1" qe a H 13 a < 121 44 : ~ 44 : 7. “I on , N 9 nN] fo. t11 8 ‘2 7 x R12 4, 7 +90 / SK Fig. 13.43 Solution to problem 13.41 (ii) The points of intersection a, b,c, d,e, f,g of the cutting plane line makes with various elements of the conein the top view. Project these points in the front view. The projection of point bof b’in the front view are to lie on 0’5’. This point cannot be plotted by direct intersection. Rotate the point b, either on 01 or on 09 (which gives the true length 0’1’ or 0’9’) and then project the point of intersection vertically into 0’1’ or 0’ in the front view and ¥’ horizontally on the 0'5' or 0'13'. (iv) Join these points in proper order and draw section lines in it. (2) To draw the true shape of the section, draw anew reference line x,y, at a convenient distance and parallel to the section plane line. Project all the points as discussed in the earlier problems. Repeat the same procedure to get the true shape of the section as shown in Fig. 13.43. PROBLEM 13.42 A right circular cone 45 mm diameter axis 65 mm long is resting on its base on HP, it is cut bya plane, the HT of which makes an angle 45° with the VP and is passing 15 mm from the top view axis. Draw the sectional front view and true shape of the section. (PTU, Jalandhar December 2003) SOLUTION. The interpretation of the solution is left to the student. See Fig. 13.44. Scanned with CamScannerIONS OF SOLIDS SECTI 317 Fig. 13.44 Solution to problem 13.42 PROBLEM 13.43 A right regular pentagonal pyramid, edge of base 25 mm, height 50 mm, lies on one of its triangular faces on HP with its axis parallel to VP. A section plane perpendicular to the HP and inclined to the VP at 30° cut the pyramid bisecting its axis. Draw its top view, sectional front view and true shape of the section, SOLUTION. (Draw the projections of the pyramid in the given position and label it. (i) Mark the mid-point of the axis in the front view and project it in the top view. Scanned with CamScanner318 ENGINEERING DRAWING AND COMPUTER GRAPHicg (iii) Draw the cutting plane line HT inclined at 30° the xy and passing through the centre of the axis. (iv) The cutting plane line HT cuts the various edges as shown in Fig. 13.45. (0) Project the points of intersection in the front view of their corresponding edges. Join these points in proper order and draw section lines in it. As the section plane is inclined to the vp, projections of the section in the front view is an apparent section. ' (vi) To draw true shape of the section, draw an auxiliary plane parallel to the section plane by the method already explained. ° WS WSS ZS > y a <| v A aa Fig. 13.45 Solution to problem 13.43 PROBLEM 13.44 A right regular pentagonal prism, side of base 25 mm and height 60 mm, rests on an edge ofits base on HP, such that one ofits base corners lies on HP and its axis is inclined at 45° to the HP and parallel tothe VP. Asection plane perpendicular to the HP and inclined tothe VP at 45° cuts the prism bisecting its axis. Draw its top view and sectional front view. SOLUTION. () Draw the projections of the prism in the given position and name it. (ii) Mark the mid-point of the axis in the front view and project it in the top view. Scanned with CamScannerSECTIONS OF SOLIDS 319 (iii) Draw the cutting plane line HT inclined at 45° axis. (jo) The cutting plane line HT cuts the various ed; (0) Project the points of intersection in the front points in proper order and draw section lines the projections of the section in the front vie to xy and passing through the centre of the Iges as shown in Fig. 13.46. view to their corresponding edges. Join these init. As the section plane is inclined to the VP, 'w is an apparent section. of | 85" Fig. 13.46 Solution to problem 13.44 PROBLEM 13.45 A vertical cylinder of 40 mm diameter is resting on its base in HP. It is cut by a section plane, which is perpendicular to the HP and inclined to the VP at 30° such that the true shape of the section is a rectangle of 30 mm x 60 mm. Draw the front view, sectional top view and true shape of the section. SOLUTION. () Draw the projections of the cylinder in the given position and name it. (i) Drawa line x,y, in such a way that the chord length in the top view is 30 mm at 30° to xy line. (iii) Project the points of intersection and draw the rectangle of 30 mm x 60 mm as shown in Fig. 13.47. Scanned with CamScanner320 ENGINEERING DRAWING AND COMPUTER GRAPHic. Fig. 13.47 Solution to problem 13.45 PROBLEM 13.46 A right egular entagonal pyraid, edge of base 25 mm and height 60 mm, rests on its bast on HP such that one of its base edges parallel to the VP. A section plane perpendicular to the HP and inclined to the VP at 30° cuts the pyramid and is 12 mm away from the axis. Draw its top view, sectional front view and trie shape of the section. SOLUTION. The interpretation of the solution is left to the student. See Fig. 13.48. Scanned with CamScannerSECTIONS OF SOLIDS 321 Fig. 13.48 Solution to problem 13.46 PROBLEM 13.47 A hexagonal pyramid, side of base 30 mm and height 60 mm, has its baseon the HP and an edge of the base makes an angle of 30° to the VP. It is cut by a plane perpendicular to the HP and inclined at 30° to VP at a distance of 18 mm from the base along the axis. Draw the sectional elevation and true shape of the section. Or A right regular hexagonal pyramid, edge of base 30 mm and height 60 mm, rests on its base in HP, with one of its base edges perpendicular to VP. A section plane inclined to VP at 30° and perpendicular to HP and is 18 ‘mm away from the axis. Draw its top view, sectional front view and true shape of the section. (PTU, Jalandhar December 2005) SOLUTION. The interpretation of the solution is left to the reader. See Fig. 13.49. Scanned with CamScanner2 eA ; TAM ro / yA /STIONS OF SOLIDS. SEC 323 SOLUTION. The solution of this problem is self-explanat tory. See Fig. 13.50. o a, oa 5 ze te 7 ba ‘ < i 7 ft ly w 1 ol le a le, 20 eS 4 > : * =a 5 a ay Fig. 13.50 Solution to problem 13.48 Scanned with CamScannerS24 ENGINEERING DRAWING AND COMPUTER GRAPHicg, PROBLEM 13.49 A right regular hexagonal prism, edge of base 30 mm and height 80 mm long has an edge ofits base in HP with its axis inclined at 60° to the HP and parallel to the VP. A section plane inclined to the VP at 60° and perpendicular to the HP cuts the axis at a distance of 60 mm from its bottoms end. Drawiits top view and sectional front view. SOLUTION. All the construction lines are retained to make the solution self-explanatory see Fig. 13.51. 4454 ea iat or icig ete ait ~ I! 2444 gy H gz q A 4,04] \ oy die’ cif bia" (ete A 6 if; 5 # 1 2 t_| ; ; a 3 i a) 3 ot Otte 701 d b ybs 2 1 F 4 z a Ws 3 oe a Fig. 13.51 Solution to problem 13.49 13.10 SECTION PLANE PERPENDICULAR TO BOTH HP AND VP When a section plane is perpendicular to both HP and VP, its HT and VT become collinear. The Projection of such a section is an edge view i. a line in both front and top views. Whereas, the projection of the section in the profile plane gives true shape of the section. PROBLEM 13.50 A right circular cone, diameter of base 50 mm, axis 50 mm long, rests on is base on HP. A section plane perpendicular to both HP and VP cuts the cone and 10 mm away from the axis. Draw its front view, top view and sectional left side view. SOLUTION. () Draw the projections of the cone in the given position and name the points on it. (i Since, the section plane is perpendicular to both HP and VP, so the section plane line is seen as an edge view ora line both in front and top views. The side view will show the true shape of the section. (iii) The cutting plane line i.e, HT and VT both cuts the various elements as shown in Fig. 13.52. (2) Project all the points of intersection in the left side view to their corresponding elements. Join these points in proper order and draw section lines in it. Scanned with CamScannergeCTIONS OF SOLIDS an z ; |. ° vw" : 7 S 7 Loss \ abs : gfeto'e F213 Tq T 397 sie" gF" gran aren 11 10 16 9 + . ° 1 8 050 4 7 7 bie —+| 10 -— H Fig. 13.52 Solution to problem 13.50 PROBLEM 13.51 A right circular cone diameter of base 50 mm and height 60 mm, rests on its base rim on HP, such that its axis is parallel to VP and inclined at 45° to the HP. A section plane perpendicular to both HP and VP, cuts the axis of the cone at a distance of 25 mm from its vertex. Draw its front view, top view and Sectional side view, SOLUTION, The interpretation of the solution is left to the student. See Fig. 13.53. Scanned with CamScannerae ENGINEERING DRAWING AND COMPUTER GRAPHicg x |] 7h \ + J n Fig, 13.53 Solution to problem 13.51 PROBLEM 13.52 A right regular pentagonal prism, side of base 25 mm, height 50 mm, lies on one ofits rectangular faces on HP, such that its axis is inclined at 30° to the VP and parallel to the HP. A section plane perpendicular to both HP and VP cuts the prism, meeting its axis at a distance of 10 mm from the base whichis ‘away from the VP. Draw its front view, top view and sectional side view. SOLUTION. (9 Draw the projections of the prism in the given position and label it. (i) As the section plane is perpendicular to both HP and VP, so its section plane line is seen as an edge view ora line both in front and top views. The side view will show the true shape of the section. (iii) The cutting plane line cuts the various elements as shown in Fig. 13.54. (iv) Projectall the points of intersection in the left side view to their corresponding edges (elements). Join these points in proper order and draw section lines in it. Scanned with CamScannerSECTIONS OF SOLIDS 50 327 y e et j ” ewe i : a No . Sol de th te Pl A & . 2" x f \ : 7 ‘ ot a a a ‘| aay 7 fen p—y i jis : b x : : ia Ne : : 30° a be ed 4 y Fig. 13.54 Solution to problem 13.52 EXERCISES SECTION PLANE PARALLEL TO THE HP 13.1 13.2 13.4 Aright regular pentagonal pyramid, side of base 25 mm and height 50 mm, rests on its base on HP such that one of its base edges is parallel to the VP. Asection plane parallel to the HP cuts the pyramid bisecting its axis. Draw its front view and sectional top view. A square pyramid, edge of base 25 mm and height 50 mm rests on its base on HP such that one of its base edges makes an angle of 30° with the VP. A section plane parallel to the HP and perpendicular to the VP cuts the pyramid bisecting its axis. Draw its front view and sectional top view. A right circular cone, diameter of base 50 mm and height 60 mm rests on HP on its base rim such that its axis is parallel to the VP and inclined to the HP at 45°. A section plane parallel to the HP and perpendicular to the VP cuts the cone bisecting its axis. Draw its front view and sectional top view. Arightregular: onal prism, side of base 25 mm and axis 50 mm, lies on one of its: mae TEP skh that sees is parallel to the both HP and VP. Asection plane parallel to and 15 mm above the HP cuts the prism. Draw its front view and sectional top view. SECTION PLANE PARALLEL TO THE VP 13.5 Accube of 35 mm edge rests on its base on HP such that one of its faces is inclined at 30° to the VP. Asection plane parallel to the VP cuts the cube at a distance of 10mm from the axis. Draw its top view and sectional front view. Scanned with CamScanner