Shape and measurement
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1 C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do we find the surfce re of shpe? Wht is line symmetry? Wht is rottionl symmetry? Wht does it men when we sy tht two figures re similr? Wht re the tests for similrity for tringles? How do we know whether two solids re similr? Wht is tesselltion nd how is it used in rt nd design? 5.1 Pythgors theorem Pythgors theorem is reltionship connecting the side lengths of right-ngled tringle. In right-ngled tringle, the side opposite the right ngle is clled the hypotenuse. The hypotenuse is lwys the longest side of right-ngled tringle. Hypotenuse Pythgors theorem Pythgors theorem sttes tht, for ny right-ngled tringle, the sum of the res of the squres of the two shorter sides ( nd ) equls the re of the squre of the hypotenuse (h). h 2 =
2 174 Essentil Stndrd Generl Mthemtics Pythgors theorem cn e used to find the length of the hypotenuse in right-ngled tringle. Exmple 1 Using Pythgors theorem to clculte the length of the hypotenuse Clculte the length of the hypotenuse in the tringle opposite, correct to 2 deciml plces. 4 cm h cm 1 Write Pythgors theorem. h 2 = Sustitute known vlues. h 2 = Tke the squre root of oth sides, then evlute. h = = Write your nswer correct to 2 deciml plces, with correct units. 10 cm The length of the hypotenuse is cm, correct to 2decimlplces. Hint: To ensure tht you get deciml nswer, set your clcultor to pproximte or deciml mode. (See the Appendix.) Pythgors theorem cn lso e rerrnged to find sides other thn the hypotenuse. Exmple 2 Using Pythgors theorem to clculte the length of n unknown side in right-ngled tringle Clculte the length of the unknown side, x, in the tringle opposite, correct to 1 deciml plce. 4.7 mm x mm 11 mm 1 Write Pythgors theorem = h 2 2 Sustitute known vlues nd the given vrile. x = Rerrnge the formul to mke x the suject, x = then evlute. = Write your nswer correct to 1 deciml plce, with correct units. The length of x is 9.9mm, correct to 1 deciml plce.
3 Pythgors theorem cn e used to solve mny prcticl prolems. Chpter 5 Shpe nd mesurement 175 Exmple 3 Using Pythgors theorem to solve prcticl prolem A helicopter hovers t height of 150 m ove the ground nd is horizontl distnce of 220 m from lnding pd. Find the direct distnce of the helicopter from the lnding pd, correct to 2 deciml plces. 1 Drw digrm to show which distnce is to e found. h m 150 m 220 m 2 Write Pythgors theorem. h 2 = Sustitute known vlues. h 2 = Tke the squre root of oth sides, then evlute. h = = Write your nswer correct to 2 deciml plces, with correct units. The helicopter is m from the lnding pd, correct to 2 deciml plces. Exercise 5A 1 Find the length of the unknown side in ech of these tringles, correct to 1 deciml plce. 54 cm c h cm 2.5 cm 4.2 cm 63.2 cm h cm 26 mm x mm 10 mm d 3.3 mm e 15.7 mm f y mm 2.3 mm 22.3 mm k mm x cm 3.9 cm 6.3 cm
4 176 Essentil Stndrd Generl Mthemtics g h i 4.5 cm v cm mm 158 mm 19.8 m t m 4.5 cm 212 mm 12.4 m 2 A frm gte tht is 1.4 m high is supported y digonl r of length 3.2 m. Find the width of the gte, correct to 1 deciml plce. 3.2 m 1.4 m 3 Aldderrestsginstrickwllsshowninthe digrm on the right. The se of the ldder is 1.5m from the wll, nd the top reches 3.5 m up the wll. Find the length of the ldder, correct to 1 deciml plce. 4 The se of ldder lening ginst wll is 1.5 m from the se of the wll. If the ldder is 5.5 m long, find how high the top of the ldder is from the ground, correct to 1 deciml plce. 3.5 m 1.5 m 5.5 m 1.5 m 5 A ship sils 42 km due west nd then 25 km due south. How fr is the ship from its strting point? (Answer correct to 2 deciml plces.) W 42 N O E 25 S 6 A ycht sils 12 km due est nd then 9 km due north. How fr is it from its strting point? 7 A hiker wlks 10 km due west nd then 8 km due north. How fr is she from her strting point? (Answer correct to 2 deciml plces.)
5 Chpter 5 Shpe nd mesurement In tringle ABC, there is right ngle t B. AB is 12 cm nd BC is 16 cm. Find the length of AC. A 12 cm B 16 cm C 9 Find, correct to 1 deciml plce, the length of the digonl of rectngle with dimensions 8.5my4m. 4 m 8.5 m 10 A rectngulr lock of lnd mesures 28 m y 55 m. John wnts to put fence long the digonl. How long will the fence e? (Answer correct to 3 deciml plces.) 11 A squre hs digonls of length 6 cm. Find the length of its sides, correct to 2 deciml plces. 6 cm 5.2 Pythgors theorem in three dimensions When solving three-dimensionl prolems, it is essentil to drw creful digrms. In generl, to find lengths in solid figures, we must first identify the correct right-ngled tringle in the plne contining the unknown side. Rememer, plne is flt surfce, such s the cover of ook or tle top. Once it hs een identified, the right-ngled tringle should e drwn seprtely from the solid figure, displying s much informtion s possile. x cm Exmple 4 Using Pythgors theorem in three dimensions The cue in the digrm on the right hs sides of length 5 cm. Find the length: AC, correct to 2 deciml plces. AD, correct to 1 deciml plce. 5 cm A 5 cm B 5 cm D C
6 178 Essentil Stndrd Generl Mthemtics 1 Locte the relevnt right-ngled tringle in the digrm. 2 Drw the right-ngled tringle ABC tht contins AC, nd mrk in known side lengths. 3 Using Pythgors theorem, clculte the length AC. 4 Write your nswer with correct units nd correct to 2 deciml plces. 1 Locte the relevnt right-ngled tringle in the digrm. 2 Drw the right-ngled tringle ACD tht contins AD nd mrk in known side lengths. (From prt, AC = 7.07 cm, correct to 2 deciml plces.) 3 Using Pythgors theorem, clculte the length AD. 4 Write your nswer with correct units nd correct to 1 deciml plce. A 5 cm C B AC 2 = AB 2 + BC 2 AC = = cm The length AC is 7.07 cm, correct to 2 deciml plces. A 7.07 cm D C AD 2 = AC 2 + CD 2 AD = = cm The length AD is 8.7cm,correct to 1 deciml plce. Exmple 5 Using Pythgors theorem in three-dimensionl prolems E For the squre pyrmid shown in the digrm, clculte: the length AC, correct to 2 deciml plces the height EF, correct to 1 deciml plce 26 cm D F C 25 cm A 25 cm B
7 Chpter 5 Shpe nd mesurement Locte the relevnt right-ngled tringle in the digrm. 2 Drw the right-ngled tringle ABC tht contins AC, nd mrk in known side lengths. 3 Using Pythgors theorem, clculte the length AC. 4 Write your nswer with correct units nd correct to 2 deciml plces. C A 25 cm B AC 2 = AB 2 + BC 2 AC = = cm The length AC is cm, correct to 2 deciml plces. 1 Locte the relevnt right-ngled tringle in the digrm. 2 Drw the right-ngled tringle EFC tht contins EF, nd mrk in known side lengths. E 26 cm 3 Find FC, which is hlf of AC. Use the vlue of AC clculted in prt. F C FC = AC 2 = = cm, correct to 2 deciml plces 4 Using Pythgors theorem, find EF. EF 2 = EC 2 FC 2 EF = = Write your nswer with correct units nd The height, EF, is 19.1cm,correctto correct to 2 deciml plces. 1decimlplce. Exercise 5B 1 The cue shown in the digrm hs sides of 3 cm. Find the length of: E H F G AC, correct to 3 deciml plces AG, correct to 2 deciml plces A 3 cm D B 3 cm C 3 cm
8 180 Essentil Stndrd Generl Mthemtics 2 For this cuoid, clculte, correct to 2 deciml plces, the length: DB BH c AH. E A D H 10 cm F B G 5 cm C 4 cm 3 Find the sloping height, s, of ech of the following cones, correct to 2 deciml plces. 84 mm s mm 25 mm s mm 96 mm 12 mm 4 The slnt height of this circulr cone is 10 cm nd the dimeter of its se is 6 cm. Clculte the height of the cone, correct to 2 deciml plces. 10 cm h cm 6 cm 5 For ech of the following squre-sed pyrmids find, correct to 1 deciml plce: i the length of the digonl on the se ii the height of the pyrmid. 10 cm 6.5 cm 6 cm 7.5 cm 7.5 cm 6 cm
9 6 Find the length of the longest pencil tht will fit inside cylinder with height 15 cm nd with circulr end surfce 8 cm in dimeter. Chpter 5 Shpe nd mesurement cm Pencil 8 8 cm cm 7 Srh wnts to put her pencils in cylindricl pencil cse. Wht is the length of the longest pencil tht would fit inside cylinder of height 12 cm with with se dimeter of 5 cm? 8 Chris wnts to use rectngulr pencil ox. Wht is the length of the longest pencil tht would fit inside the ox shown on the right? (Answer to the nerest centimetre.) 20 cm 12 cm 5.3 Mensurtion: perimeter nd re Mensurtion is prt of mthemtics tht looks t the mesurement of length, re nd volume. It comes from the Ltin word mensur, which mens mesure. Perimeter The perimeter of two-dimensionl shpe is the distnce round its edge. 10 cm Exmple 6 Finding the perimeter of shpe Find the perimeter of the shpe shown. 16 cm 12 cm To find the perimeter, dd up ll the side lengths of the shpe. 25 cm Perimeter = = 81 cm
10 182 Essentil Stndrd Generl Mthemtics Are The re of shpe is mesure of the region enclosed y the oundries of the shpe. When clculting re, the nswer will e in squre units, i.e. mm 2, cm 2, m 2, km 2, etc. The formuls for the res of some common shpes re given in the tle elow, long with the formul for finding the perimeter of rectngle. Rectngle Shpe Are Perimeter l w A = lw P = 2l + 2w or P = 2(l + w) Prllelogrm Sum of four sides h A = h Trpezium h A = 1 ( + )h 2 Sum of four sides Tringle Sum of three sides A = 1 2 h Exmple 7 Finding the perimeter of rectngle Find the perimeter of the rectngle shown. 5 cm 12 cm
11 Chpter 5 Shpe nd mesurement Since the shpe is rectngle, use the formul P = 2l + 2w. P = 2L + 2W 2 Sustitute length nd width vlues into = the formul. 3 Evlute. = 34 cm 4 Give your nswer with correct units. The perimeter of the rectngle is 34 cm. Exmple 8 Finding the re of shpe Find the re of the given shpe. 4.9 cm 5.8 cm 7.6 cm 1 Since the shpe is trpezium, use the formul A = 1 ( + )h. 2 A = 1 ( + )h 2 2 Sustitute the vlues for, nd h. = 1 ( ) Evlute. = cm 2 4 Give your nswer with correct units. The re of the shpe is cm 2. You cn use the formuls to find re nd perimeter in prcticl prolems. Exmple 9 Finding the perimeter nd re in prcticl prolem A tle is to e covered in tiles with n edging round the perimeter. The tle mesures 150 cm y 90 cm. Wht length of edging is required? Wht re will e covered with tiles? 1 To find the length of edging, we need to work out the perimeter of the tle. Since the tle is rectngle, use the formul P = 2l + 2w.
12 184 Essentil Stndrd Generl Mthemtics 2 Sustitute l = 150 nd w = 90 nd evlute. P = 2L + 2W = 2(150) + 2(90) = 480 cm 3 Give your nswer with correct units. The length of edging required is 480 cm. 1 To find the re, use the formul A = lw. A = LW 2 Sustitute l = 150 nd w = 90 nd evlute. = = cm 2 3 Give your nswer with correct units. The re to e covered with tiles is cm 2. Exercise 5C 1 For ech of the following shpes, find, correct to 1 deciml plce: i the perimeter ii the re 15 cm 3.3 cm 7.9 cm c d 78 cm 130 cm 5 cm 7 cm 104 cm 15 cm 2 Find the res of the given shpes, correct to 1 deciml plce, where pproprite. 4.2 m c 15.2 m 7.4 m 3.7 m 4.8 m 6.6 cm 15.7 cm d e 9.5 cm f 9.4 cm 2 cm 15.7 cm 4.5 cm 10.4 cm 2 cm 6.9 cm
13 Chpter 5 Shpe nd mesurement 185 g h 5 m 8.8 m 1.8 m 25 m 12.5 m 3 A 50 m swimming pool increses in depth from 1.5 m t the shllow end to 2.5 m t the deep end, s shown in the digrm (not to scle). Clculte the re of side wll of the pool. 50 m 1.5 m 2.5 m 4 Admwllisuiltcrossvlleythtis550mwidetitssend1.4 km wide t its top, s shown in the digrm (not to scle). The wll is 65 m deep. Clculte the re of the dm wll. 1.4 km 65 m 550 m 5 Ry wnts to tile rectngulr re mesuring 1.6 m y 4 m outside his holidy house. The tiles tht he wishes to use re 40 cm y 40 cm. How mny tiles will he need? 6 One litre of pint covers 12 m 2. How much pint is needed to pint wll mesuring 3 m y 12 m? 5.4 Circles The perimeter of circle is known s the circumference (C) of the circle. Rdius Circumference Dimeter
14 186 Essentil Stndrd Generl Mthemtics The re nd the circumference of circle re given y the following formule. Are Circumference Circle r A = r 2 where r is the rdius C = 2 r or C = d where d is the dimeter Exmple 10 Finding the circumference nd re of circle For the circle shown, find: the circumference, correct to 1 deciml plce the re, correct to 1 deciml plce 3.8 cm 1 For the circumference, use the formul C = 2 r. C = 2πr 2 Sustitute r = 3.8 nd evlute. = 2π 3.8 = Give your nswer correct to 1 deciml plce nd with correct units. 1 To find the re of the circle, use the formul A = r 2. A = πr 2 2 Sustitute r = 3.8 nd evlute. = π Give your nswer correct to 1 deciml plce nd with correct units. The circumference of the circle is 23.9cm,correctto1decimlplce. = The re of the circle is 45.4cm 2, correct to 1 deciml plce. Exercise 5D 1 For ech of the following circles, find: i the circumference, correct to 1 deciml plce ii the re, correct to 1 deciml plce
15 Chpter 5 Shpe nd mesurement 187 c 5 cm 8.5 cm 4.08 m d e f 15.8 mm 15 cm 0.4 m 2 For ech of the following shpes, find: i the perimeter, correct to 2 deciml plces 10 cm ii the re, correct to 2 deciml plces. 28 mm 495 mm c d 8 mm 57 cm 3 Find the shded res in the following digrms, correct to 1 deciml plce. 4.8 cm 11.5 cm m
16 188 Essentil Stndrd Generl Mthemtics c d 4.2 cm 8.7 cm 350 mm 350 mm 430 mm 7.6 cm 860 mm 4 Afenceneedstoeuiltroundnthleticstrckththsstrights400mlongnd semicirculr ends of dimeter 80 m. Correct to 2 deciml plces: wht length of fencing is required? wht re will e enclosed y the fencing? 5.5 Volume Volume is the mount of spce occupied y three-dimensionl oject. The volume of prism or cylinder is found y using its cross-sectionl re. Prisms nd cylinders re three-dimensionl ojects tht hve uniform cross-section long their entire length. Cross-section Height Length For prisms nd cylinders: Volume = re of cross-section height (or length) When clculting volume, the nswer will e in cuic units, i.e. mm 3, cm 3,m 3, etc. The formuls for the volumes of regulr prisms nd cylinder re given in the tle elow. Shpe Rectngulr prism (cuoid) Volume h V = lwh w l
17 Chpter 5 Shpe nd mesurement 189 Squre prism (cue) l l l V = l 3 Tringulr prism h V = 1 2 hl l Cylinder r h V = r 2 h Exmple 11 Finding the volume of cuoid Find the volume of the following cuoid. 4 cm 6 cm 12 cm 1 Use the formul V = lwh. V = LWH 2 Sustitute in l = 12, w = 6 nd h = 4. = Evlute. = 288 cm 3 4 Give your nswer with correct units. The volume of the cuoid is 288 cm 3. Exmple 12 Finding the volume of cylinder Find the volume of this cylinder. Give your nswer correct to 2 deciml plces. 5 m 25 m
18 190 Essentil Stndrd Generl Mthemtics 1 Use the formul V = r 2 h. V = πr 2 h. 2 Sustitute in r = 5 nd h = 25 nd evlute. = π = Write your nswer correct to 2 deciml plces nd with correct units. The volume of the cylinder is cm 3, correct to 2 deciml plces. Exmple 13 Finding the volume of three-dimensionl shpe Find the volume of the three-dimensionl shpe shown. 10 cm 6 cm 25 cm 15 cm Strtegy: To find the volume, find the re of the shded cross-section nd multiply it y the length of the shpe. 1 Find the re of the cross-section, which is trpezium. Use the formul A = 1 2 ( + )h. Are of trpezium = Sustitute in = 10, = 15 nd h = 6 nd evlute. 2 To find the volume, multiply the re of the cross-section y the length of the shpe (25 cm). 1 ( + )h 2 = 1 ( )6 2 = 75 cm 2 V = re of cross-section length = = 1875 cm 3 3 Give your nswer with correct units. The volume of the shpe is 1875 cm 3.
19 Chpter 5 Shpe nd mesurement 191 Exercise 5E 1 Find the volumes of the following solids. Give your nswers correct to 1 deciml plce where pproprite. c 5 cm 35 cm 51 cm 18.9 cm 15.6 cm 12.5 cm d e 20 cm f 12.7 mm 35.8 mm 10 cm 58 cm 75 cm 2.5 m 0.5 m g 14 mm h 3.8 m 0.48 m 11.8 cm 13.5 cm 2 m 2 A cylindricl plstic continer is 15 cm high nd its circulr end surfces ech hve rdius of 3 cm. Wht is its volume, to the nerest cm 3? 3 Wht is the volume, to the nerest cm 3, of rectngulr ox with dimensions 5.5 cm y 7.5 cm y 12.5 cm? 4 Wht is the volume, correct to 2 deciml plces, of cylindricl pint tin with height 33 cm nd dimeter 28 cm? Given tht there is 1 millilitre (ml) in 1 cm 3, how mny litres of pint would fill this pint tin? Give your nswer to the nerest litre. 5 Find the volume of n equilterl tringulr prism with height 12 cm nd side length 2 cm. Give your nswer correct to 3 deciml plces. 12 cm 2 cm
20 192 Essentil Stndrd Generl Mthemtics 5.6 Volume of cone A cone cn fit inside cylinder, s shown in the digrm. The cone occupies one-third of the volume of the cylinder contining it. The formul for finding the volume of cone is, therefore: Volume of cone = 1 volume of its cylinder 3 Volume of cone = 1 re of se height 3 = 1 3 r 2 h Exmple 14 Finding the volume of cone Find the volume of this right circulr cone. Give your nswer correct to 2 deciml plces. 15 cm 1 Use the formul for the volume of cone, V = 1 3 r 2 h. V = 1 3 πr 2 h 8.4 cm 2 Sustitute r = 8.4 nd h = 15 nd evlute. = 1 3 π(8.4)2 15 = Give your nswer correct to 2 deciml plces nd with correct units. The volume of the cone is cm 3,correct to 2 deciml plces. Exercise 5F 1 Find the volume of these cones, correct to 2 deciml plces m 28 cm 2.50 m 18 cm
21 Chpter 5 Shpe nd mesurement 193 c d 1 m 755 cm 40 mm 15 mm 2 Find the volume (correct to 2 deciml plces) of the cones with the following dimensions. Bse rdius 3.50 cm, height 12 cm Bse rdius 7.90 m, height m c Bse dimeter 6.60 cm, height 9.03 cm d Bse dimeter cm, height cm 3 Wht volume of crushed ice will fill snow cone level to the top if the snow cone hs top rdius of 5 cm nd height of 15 cm? Answer correct to 3 deciml plces. 5.7 Volume of pyrmid A squre pyrmid cn fit inside prism, s shown in the digrm. The pyrmid occupies one third of the volume of the prism contining it. The formul for finding the volume of pyrmid is therefore: Volume of pyrmid = 1 volume of its prism 3 Volume of pyrmid = 1 re of se height 3 = 1 3 lwh Exmple 15 Finding the volume of squre pyrmid Find the volume of this squre right pyrmid of height 11.2 cm nd se 17.5 cm. Give your nswer correct to 2 deciml plces cm 17.5 cm 17.5 cm
22 194 Essentil Stndrd Generl Mthemtics 1 Use the formul V = 1 re of se height 3 V 2 Sustitute the vlues for the re of the se (in this exmple, the se is squre) nd height of the pyrmid nd evlute. 3 Give your nswer correct to 2 deciml plces nd with correct units. = 1 re of se height 3 = = The volume of the pyrmid is cm 3,correctto2deciml plces. Exmple 16 Finding the volume of hexgonl pyrmid Find the volume of this hexgonl pyrmid. 12 cm 1 Use the formul V = 1 re of se height 3 V 2 Sustitute the vlues for re of se (122 cm 2 ) nd height (12 cm) nd evlute. Are of se = 122 cm 2 = 1 re of se height 3 = = 488cm 3 3 Give your nswer with correct units. The volume of the pyrmid is 488 cm 3. Exercise 5G 1 Find the volumes of the following right pyrmids, correct to 2 deciml plces where pproprite. 5 cm 15 m 4 cm 12 m 7 m
23 Chpter 5 Shpe nd mesurement 195 c d 4.5 m 4.56 cm Are of se = 16 m cm 2 A squre-sed pyrmid hs se side length of 8 cm nd height of 10 cm. Wht is its volume? Answer correct to 3 deciml plces. 3 The first true pyrmid in Egypt is known s the Red Pyrmid. It hs squre se pproximtely 220 m long nd is out 105 m high. Wht is its volume? 4 Find the volumes of these composite ojects, correct to 1 deciml plce. 5.8 cm 5.8 cm 3.2 cm 8.5 cm 3.2 cm 3.5 cm 5.8 Volume of sphere The volume of sphere cn e found y using the formul: V = 4 3 r 3 where r is the rdius of the sphere. r
24 196 Essentil Stndrd Generl Mthemtics Exmple 17 Finding the volume of sphere Find the volume of this sphere, giving your nswer correct to 2 deciml plces cm 1 Use the formul V = 4 3 r 3. V = 4 3 πr 3 2 Sustitute r = 2.5 nd evlute. = 4 3 π Give your nswer correct to 2 deciml plces nd with correct units. = The volume of the sphere is cm 3,correctto2deciml plces. Exercise 5H 1 Find the volumes of these spheres, giving your nswers correct to 2 deciml plces. c 24 cm 5 mm 3.8 mm 2 Find the volumes, correct to 2 deciml plces, of the following lls: tennis ll, rdius 3.5cm sketll, rdius 14 cm c Golf ll, rdius, 2 cm 3 Find the volumes, correct to 2 deciml plces, of the following hemispheres. 16 cm c 10 cm d cm 2.5 m
25 Chpter 5 Shpe nd mesurement Lois wnts to serve punch t Christms time in her new hemisphericl owl with dimeter of 38 cm. How mny litres of punch could e served, given tht 1 millilitre (ml) is the mount of fluid tht fills 1 cm 3? Answer to the nerest litre. 5.9 Surfce re To find the surfce re (SA) of solid, you need to find the re of ech of the surfces of the solid nd then dd these ll together. Solids with plne fces (prisms nd pyrmids) It is often useful to drw the net of solid to ensure tht ll sides hve een dded. Exmple 18 Finding the surfce re of pyrmid Find the surfce re of this squre-sed pyrmid. 6 cm 1 Drw net of the squre pyrmid. 8 cm 8 cm 6 cm 2 Write down the formul for totl surfce re, using the net s guide. Totl surfce re = re of squre se + 4 re of tringle
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