Geometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm

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1 Reding Scles There re two things to do when reding scle. 1. Mke sure you know wht ech division on the scle represents. 2. Mke sure you red in the right direction. Mesure Length metres (m), kilometres (km), centimetres (cm) Weight grm (g), kilogrms (kg), tonne Volume/Cpcity litre (l), centilitre (cl), millilitre (ml) Time yers, weeks, dys, hours nd minutes Time This clock hs two hnds The short hnd tells us the hour nd the long hnd tells us the minutes. For the hour hnd ech mrk is one hour. For the minute hnd ech mrk is 5 minutes. This clock reds 20 pst 8. We write this s 8:20. Times efore noon hve.m. eside them Times fter noon hve p.m. 7:45 t night is 7:45 p.m. 8:20 in the morning is 8:20.m. 24 Hour Time Twenty pst eight could e in the morning or t night The 24 hour clock numers the hours fter middy s 13, 14, 15, m 1m 2m 3m 4m 5m 6m 7m 8m 9m 10m 11m 12pm midnight noon fternoon nd Evening pm 1pm 2pm 3pm 4pm 5pm 6pm 7pm 8pm 9pm 10pm 11pm 12pm noon midnight 3:20 pm is written s 1520 hours on 24 hour clock. We sy this s fifteen twenty hours Twenty four hour clock times lwys hve 4 digits. 8:20 m is written s Geometry nd Mesure GCSE Grde G Time Mesure There re 60 seconds in 1 minute There re 60 minutes in 1 hour There re 24 hours in 1 dy There re 30 dys in pril, June, Septemer nd Novemer There re 31 dys in Jnury, Mrch, My, July, ugust, Octoer nd Decemer. There re 28 dys in Ferury if the yer is not lep yer. There re 29 dys in Ferury if the yer is lep yer. There re 7 dys in week There re 52 weeks in yer There re 12 months in 1 yer. Time Mesures When dding or sutrcting time, it is est to dd or sutrct the hours nd minutes seprtely. On Sturdy, Dle prctised the guitr from m. until 1 p.m. nd then from 4.25 p.m. until 5.15 p.m. For how long did Dle prctise on Sturdy? From m. until 1 p.m. is 1 hour 20 minutes. From p.m. until 5.15 p.m. is 35 minutes + 15 minutes. Totl hours = 1 hour Totl minutes = = 70 minutes = 1 hour 10 minutes. Totl time = 1 hour + 1 hour 10 minutes = 2 hours 10 minutes. Types of ngle right ngle is ¼ turn. stright ngle is ½ turn. n cute ngle is smller thn right ngle. n otuse ngle is greter thn right ngle ut smller thn stright ngle. reflex ngle is greter thn stright ngle ut less thn complete turn. There re 90 in ¼ turn. turn of 45 is ½ of ¼ turn turn of 30 is 1/3 of ¼ turn. turn of 60 is 2/3 of ¼ turn Perpendiculr nd Prllel Lines This line is clled since it runs from to. It cn lso e clled since it runs from to. D Prllel lines re lwys the sme distnce prt. Prllel lines never meet. C In this digrm the line is prllel to the line CD. We write //CD. The symol // is red s is prllel to We often put rrows on lines to show they re prllel. C Perpendiculr lines re t right ngles to ech other. In the digrm the line is perpendiculr to the line CD. We write CD. The symol is red s is perpendiculr to D

2 Perimeter The distnce round the outside of shpe is clled the perimeter Perimeter is mesured in mm, cm, m, or km. Find the perimeter of this rectngle. 7 cm 3 cm Perimeter = = 20 cm For Grde F the perimeter of compound shpe would hve to e found. Find the perimeter of this compound shpe. 8cm The lengths of the two missing sides re 6 cm nd 4 cm. So the perimeter = = 28 cm re 6cm The mount of surfce shpe covers is clled its re. This squre is 1cm long nd 1cm wide. The re of this squre is clled 1 squre centimetre. We write 1 squre centimetre s 1cm 2 Some units for mesuring re re mm 2, cm 2, m 2, km 2. Volume The volume of shpe is the mount of spce it tkes up 1 cuic centimetre is the mount of spce this cue tkes up. 1 cuic centimetre is written 1cm 3. We sy tht the volume of cue of side 1cm is 1cm 3. Units of volume re mm 3, cm 3, m 3 Specil Tringles nd Qudrilterls Nming Polygons 3-sided polygon is tringle 4 sided polygon is qudrilterl 5-sided polygon is pentgon 6 sided polygon is hexgon 7-sided polygon is heptgon n 8 sided polygon is octgon 9-sided polygon is nongon 10 sided polygon is decgon regulr polygon hs ll its sides equl nd ll its ngles equl. Edge Fces Edges Vertices fce is flt surfce. This shpe hs 6 fces. n edge is line where two fces meet. This shpe hs 12 edges. vertex is corner where edges meet (Vertices is the plurl of vertex). This shpe hs 8 vertices. Digonl Lines The lines drwn from one corner to the opposite corner re clled digonls Circles circle is drwn ccurtely using compss. The line from the centre of the circle to the circumference is clled the rdius The dimeter of the circle is the width. Rdius Dimeter Circumference Fce Congruence Identicl shpes re known s congruent shpes. If trcing of one shpe will fit exctly over nother shpe, the shpes re sid to e congruent Congruent shpes re the sme shpe nd size. These shpes re congruent Congruent shpes: o re the sme size; o hve ngles of the sme size; o hve sides of the sme length; Congruent solids hve ngles nd sides nd fces of the sme size. Vertex Equilterl Isosceles Right-ngled Sclene n equilterl tringle hs ll its sides equl nd ll its ngles equl. n isosceles tringle hs two equl sides nd two equl ngles. The equl ngles re opposite the equl sides. right-ngled tringle hs one ngle equl to 90. sclene tringle hs not equl sides nd no equl ngles.

3 Estimting Sometimes when we estimte it is etter to overestimte nd sometimes it is etter to underestimte Overestimte mount of pint for room Underestimte numer of people in lift. If we know the height of something we cn estimte the height or length of other ojects. Metric Units Length Conversions For converting etween the commonly used units for length the reltionships in this list cn e used 1km = 1000m 1m = 1000mm 1m = 100cm 1cm = 10mm To chnge smll units to lrger units, lwys divide. 732 cm to metres = 7.32 m To chnge lrge units to smller units, lwys multiply. Chnge 1.2 m to centimetres Weight Conversions 1kg = 1000g 1tonne = 1000kg 1.2 x 100 = 120 cm To chnge smll units to lrger units, lwys divide. To chnge lrge units to smller units, lwys multiply. Cpcity Conversions 1cl = 10ml 100cl = 1000ml To chnge smll units to lrger units, lwys divide. To chnge lrge units to smller units, lwys multiply. Volume 1000 litres = 1 m 3 1 ml = 1 cm 3 ngles Estimting ngle Size The est estimte you would e expected to give would e to the nerest 10. When estimting the size of n ngle it is often helpful to compre with right ngle. Mesuring ngles We mesure the size of n ngle with protrctor. There re two sorts of protrctor; circulr one nd semicirculr one. Follow these steps to mesure n ngle using the protrctor. Step 1 Step 2 Step 3 Estimte the size of the ngle. Plce the centre of the protrctor on the vertex of the ngle. Keeping the centre on the vertex, move the protrctor round until the se line lies long one of the rms of the ngle. GCSE Grde F Step 4 Step 5 Step 6 Decide whether to use the inside or outside scle. The scle to use is the one tht hs 0 on the rm of the ngle. Red off the numer where the other rm of the ngle meets the chosen scle. Check this numer with your estimte to mke sure you hve red the correct scle. The size of n ngle which is greter thn hlf circle cn e red directly from circulr protrctor. If you hve semicirculr protrctor, then tke the following steps. Step 1 Mesure the inside ngle. Step 2 Sutrct this mesurement from 360 Drwing ngles Step 1 Drw stright line. This will e one of the rms of the ngle. Step 2 Plce the protrctor so tht the se line lies long the drwn line nd the centre of the protrctor is on the end of the drwn line. Step 3 Red round the scle tht egins 0 on the drwn line. Put smll mrk eside 130 (for exmple). Step 4 Tke the protrctor wy. Through the smll mrk, drw the other rm of the ngle. ngles of ny size etween 0 nd 180 re s ove. Using circulr protrctor, ngles of ny size my e drwn. Using semicirculr protrctor, ngles etween 180 nd 360 re s follows. Step 1 Sutrct the size of the ngle to e drwn from 360 Step 2 Drw this smller ngle. The other ngle in the digrm will now e the ngle required. Nming ngles n ngle cn e nmed y the cpitl letter t the vertex. The mrked ngle cn e nmed s the ngle Q or s Q. The symol is the symol for ngle. R n ngle cn lso e nmed y using three cpitl letters. The mrked ngle could e nmed s PQR or RQP. Q When we use three cpitl letters to nme n ngle, the first is the letter t the end of one of the rms, the middle letter is the letter t the vertex nd the lst letter is the letter P t the end of the other rm. We cnnot nme the mrked ngle s Q since there is more thn one ngle t Q. There is PQS nd RQP tking these R two ngles together there is RQS. Whenever there is more thn one ngle t point we must use three letters to nme the ngle we wnt. Q lwys rememer tht when n ngle is nmed y three cpitl letters, it is the middle letter tht is the vertex. P S

4 ngle Fcts The ngles on stright line dd up to 180 (clled the scle). : this drwing hs scle of 1:10, so nything drwn with the size of 1 would hve size of 10 in the rel world, so mesurement of 150mm on the drwing would e 1500mm on the rel horse. ngles such s,, c nd d re clled ngles t point. They re the ngles round point. c d ngles t point dd up 360 Lines of Symmetry line of symmetry is line tht cn e drwn through shpe so tht wht cn e seen on one side of the line is the mirror imge of wht is on the other side. This is why line of symmetry is sometimes clled mirror line. It is lso the line long which shpe cn e folded exctly. Find the numer of lines of symmetry for this cross. This cross hs totl of four lines of symmetry. Rel Horse Drwn Horse 1500 mm high 150mm high Nets net is flt shpe tht cn e folded into 3D shpe. The Net Completed ox re of Rectngle re = length x width or re = se x height width height Rottionl Symmetry length se The order of rottionl symmetry is the numer of times shpe fits exctly onto itself during one complete turn Since ll shpes will fit onto themselves t lest once during complete turn ll shpes hve order of rottionl symmetry of t lest 1 2D shpe hs rottionl symmetry if it cn e rotted out point to look exctly the sme in new position. Clculte the re of this rectngle. re of rectngle = length x width = 11 cm x 4 cm = 44 cm 2 11 cm 4 cm The esiest wy to find the order of rottionl symmetry for ny shpe is to trce it nd count the numer of times tht the shpe stys the sme s you turn the trcing pper through one complete turn. Find the order of rottionl symmetry for this shpe. First hold the trcing pper on top of the shpe nd trce the shpe. Then rotte the trcing pper nd count the numer of times the trcing pper mtches the originl shpe in one complete turn. You will find three different positions. So, the order of rottionl symmetry for the shpe is 3. Scle Drwings scle drwing shows rel oject with ccurte sizes except they hve ll een reduced or enlrged y certin mount For Grde D the re of compound shpe would hve to e found. Find the re of this compound shpe. 8cm First split the shpe into two rectngles, nd. Then clculte the re of ech one. re of = 2 x 6 = 12 cm 2 re of = 6 x 2 = 12 cm 2 6cm The re of the shpe is given y: re of + re of = = 24 cm 2 8cm 6cm

5 Imperil Units Imperil Mesures for Length, Weight nd Cpcity Length inch ( ), foot ( ), yrd, mile Mss ounce (oz), pound (l), stone, ton Cpcity pint, gllon Length 1 foot = 12 inches Weight 1 l = 16 oz Cpcity 1 yrd = 3 feet 1 stone = 14 pounds 1 mile = 1760 yrds 1 ton = 2240 pounds 1 gllon = 8 pints To chnge smll units to lrger units, lwys divide. To chnge lrge units to smller units, lwys multiply. Write 5 4 in inches 5 feet = 12 x 5 inches = 60 inches Then 5 4 = 60 inches + 4 inches = 64 inches When dding, mesurements which re given in mixed units egin y dding the units seprtely. The sme pplies when sutrcting or multiplying. When dividing, it is wise to convert the mesurement so tht it is given in the smller unit. recipe for mking pple strudel uses 2l 12 oz of pples. Michel ws using this recipe to mke strudel for his fmily. He ws going to doule the recipe. Wht quntity of pples should he use? Michel needs 2 x 2 l 12 oz of pples. 2 x 2 l = 4 l 2 x 12 oz = 24 oz = 1 l 8 oz So Michel needs 4 l + 1 l 8 oz = 5 l 8 oz of pples. Mss Equivlents 1kg is out 2 l 1 l is out 0.5kg or 500g recipe for fudge cke uses 8oz of utter. out how mny grms is this? 1 l of utter is out 500g. 8oz is ½ l. ½ l of utter is out ½ x 500g or 250g Cpcity Equivlents 1 Gllon is out 4.5 litres 1 litre is out 1.75 pints tnk holds 40 gllons of wter. out how mny litres is this? GCSE Grde E 1 gllon is out 4.5 litres. 40 gllons is out 40 x 4.5 litres or 180 litres. Tht is, the tnk holds out 180 litres of wter. Length Equivlents 5 mile is out 8km 3 feet is little less thn 1 m 1 inch is out 2.5 cm Dee wlked 20km. out how mny miles did Dee wlk? Firstly, find how mny lots of 8km Dee wlked. We do this y dividing 20 y 8. We get 2.5 Since there re 5 miles in ech lot of 8km, we now multiply y 5 to find the totl numer of miles. We get 5 x 2.5 = 12.5 miles. Tht is, Dee wlked out 12 ½ miles. Tringles nd ngles The ngles,, c, re clled the interior ngles of tringle. They re the ngles inside the tringle. The sum of the interior ngles of tringle is 180 Whenever we use the fct tht the ngles inside tringle dd to 180, c we must sy so, we my use sum of Find the vlue of n 118 n 30 n = 180 ( sum of ) n = 180 n = 32 (sutrcting 148 from oth sides) ngles in Specil Tringles n equilterl tringle hs ll its sides equl nd ll its ngles equl. Ech ngle is 60 n isosceles tringle hs two equl sides nd two equl ngles. right-ngled tringle hs n interior ngle of 90 o

6 Surfce re nd Volume of Cuoid cuoid is ox shpe, ll six fces re rectngles. The volume of cuoid is given y the formul: Volume = length x width x height The surfce re of cuoid is clculted y finding the totl re of the six fces, which re rectngles. The opposite rectngles hve the sme re. The re of the top nd ottom re the sme. The re of the front nd ck re the sme. The re of the two sides rectngles re the sme. Surfce re of cuoid is given y the formul: Surfce re = 2lw + 2hw + 2hl Clculte the volume nd surfce re of this cuoid V = 5 x 3 x 4 = 60cm 3 S. re = (2 x 3 x 5) + ( 2 x 4 x 5) + ( 2 x 4 x 3) = = 94 cm 2 For Grde D the volume of compound shpe would hve to e found. Tesselltions repeting pttern of identicl shpes which fit together exctly, leving no gps. 5cm Width (w) 4cm 3cm Height (h) Length (l) Reflection Reflective symmetry cn e descried y giving the loction of the lines of symmetry. Lines of symmetry re often clled xes of symmetry. On the grid, reflect tringle P in the y-xis. The gry tringle is the originl shpe nd the red tringle is the imge. y P O x

7 Compss Directions N 60 E mens 60 Est of North To find this direction follow these steps. Step 1 Step 2 erings Fce North Turn 60 towrds Est. The ering of point from point is the ngle through which you turn clockwise s you chnge direction from due north to the direction of. For exmple in the digrm ove the ering is lwys mesure from North. 2. lwys move in clockwise direction. 3. lwys give the nswer in three figures. Intersecting nd Prllel Lines ngles su s nd re clled verticlly opposite ngles. They re the two ngles which re opposite one nother (not eside ech other) when two lines intersect. Verticlly opposite ngles re equl ngles such s nd re clled corresponding ngles. Corresponding ngles re in corresponding positions. One corresponding ngles cn e trnslted onto the other. They re sometimes clled F ngles. Corresponding ngles re equl ngles such s nd re clled lternte ngles. lternte ngles re oth inside the prllel lines nd on opposite sides of the trnsversl. One lternte ngle cn e rotted onto nother lternte ngle. They re sometimes clled Z ngles. lternte ngles re equl N 60 N 60 E = = GCSE Grde D Specil Qudrilterls Prllelogrm Opposites sides re prllel Opposite sides re equl Digonls isect ech other Opposite ngles re equl Rhomus rhomus is prllelogrm which hs ll sides equl. Its digonls isect ech other t right ngles. Its digonls lso isect the ngles Kite kite is qudrilterl with two pirs of equl djcent sides. Its longer digonl isects its shorter digonl t right ngles. The opposite ngles etween the sides of different lengths re equl. Trpezium trpezium hs two prllel sides. Using Isometric Pper to Drw 3-D Shpes Isometric mens sme mesure On n isometric drwing of 3-D shpe the lengths which re equl on the shpe re lso equl on the drwing. n isometric drwing of 3-D shpe will mke the shpe pper slightly distorted. To mke n isometric drwing of 3-D shpe we cn use either isometric grph pper or isometric dot pper. Verticl edges of the 3-D shpe should e drwn s verticl lines. Three steps to drwing cue using isometric pper ngles such s nd re clled interior ngles. Interior ngles re oth inside the prllel lines nd on the sme side of the trnsversl. Interior ngles dd to = 180 There is often more thn one wy of finding n unknown ngle. We cn use ny of verticlly opposite ngles djcent ngles on stright line ngles t point corresponding ngles lternte ngles interior ngles elow re two drwings of the sme cuoid on different isometric grids

8 re of Tringle re = se x height 2 height Find the re of this right-ngled tringle. re = ½ x 7 x 4 = 14 cm 2 se 4 cm height se 7 cm re of Prllelogrm re of prllelogrm = se x height h = x h Find the re of this prllelogrm 6 cm re = 8 x 6 = 48 cm 2 re of Trpezium 8 cm re of trpezium = hlf the sum of the prllel sides x height, the height is the perpendiculr distnce etween the prllel sides. = ½ ( + ) x h Find the re of the trpezium re = ½ (4 + 7) x 3 = 16.5 cm 2 Interior ngles of Polygon 4 cm 3 cm 7 cm To find the sum of the ngles in polygon we cn proceed s follows Step 1 From one vertex, drw ll the digonls to divide the polygon into tringles. Step 2 Find the sum of the ngles in ll of these tringles. Find the sum of the ngles in this polygon. h Trnsltions trnsltion is the movement of shpe from one position to nother without reflecting it or rotting it. It is sometime clled sliding trnsformtion, since the shpe ppers to slide from one position to nother. Descrie the trnsltion of the red rrow to the grey rrow. The rrow hs een trnslted 5 squres right nd 1 squre down. trnsltion cn lso e descried y using vector. (Grde C) vector is written in the form where descries the horizontl movement (x-direction) nd descries the verticl movement (y-direction) Giving the nswer in vector form for the exmple ove is 5 1. Rottion rottion trnsforms 2D shpe to new position y turning it out fixed point, clled the centre of rottion. The oject (grey shpe) hs een rotted 90 0 (or ¼ turn) clockwise to give the imge (red shpe) Enlrgement Scle Fctor of n enlrgement If shpe is enlrged so tht ech length ecomes twice the size it ws on the originl we sy the scle fctor of the enlrgement is 2. Shpe CD enlrged to the shpe C D Ech length on the enlrgement C D is twice s long s the corresponding length on the originl CD. The scle fctor for this enlrgement is 2 D Oject Imge D C C This polygon cn e divided into 4 tringles. Sum of the ngles in 4 tringles = 4 x 180 = Clculte Missing ngle in Polygon Work out the size of the ngle mrked. The sum of the ngles in hexgon totl 720 0, (see ove). Totl the ngles given nd sutrct from ( ) = o 117 o 98 o 134 o The tringle C hs een enlrged to the tringle C. The scle fctor of the enlrgement is 3. If you join, nd C C, the point P, where these lines meet is known s the centre of enlrgement. Drwing Enlrgements To drw n enlrgement of shpe we need to know oth the scle fctor nd the centre of enlrgement. We need the scle fctor to mke the enlrgement the right size. We need the centre of enlrgement to position the enlrgement correctly. P C C 104 o

9 Circumference nd re of Circle C = πd We cn use the formul C = πd to clculte the circumference of circle Rememer r is the rdius of circle. d = 2r Replcing d y 2r in the formul C = πd we get C = 2πr Find the circumference of these circles.. 45mm 4 cm. C = 2πr. C = πd C = 2 x 3.1 x 45 C = 3.1 x 4 C = 279mm C = 12.4cm The re of circle cn e clculted using the formul = πr 2. Find the re of these circles Tke π = 3.1 Give the nswers to 1 d.p mm 4.3cm = πr 2 r = 39.5mm = 3.1 x = πr 2 = 57.3 cm 2 (1d.p.)d = 79mm = 3.1 x = mm 2 (1d.p.) Prisms prism is 3D shpe tht hs the sme cross-section running ll the wy through it, whenever it is cut perpendiculr to its length. Tringulr prism Cuoid Cylinder Hexgonl prism Cross-section Cross-section Cross-section Cross-section isosceles tringle rectngle circle regulr hexgon The volume of prism is found y multiplying the re of its cross-section y the length of the prism. GCSE Grde C Exterior ngles of Polygon The exterior ngles of ny sided polygon totl 360. e i e i i e i e The digrm shows prt of regulr octgon. Work out the size of ngle x. Knowing tht the exterior ngles sum to 360, = 45 x Digrm NOT ccurtely drwn nd knowing djcent ngles on stright line dd to = 135 x = 135 Pythgors Theorem Pythgors Theorem gives the reltionship etween the lengths of the sides in right-ngled tringle. c c 2 = is Pythgors Theorem in symols for the tringle shown Using Pythgors Theorem We cn use Pythgors Theorem to find the length of the third side of right-ngled tringle, if we know the lengths of the other two sides. Find the vlue of x in ech of the tringles... 7cm x 15cm 9cm 9cm x. x 2 = (Pythgors Theorem) = = 130 x = 130 = 11.4cm (to 1d.p.) = x (Pythgors Theorem) Rewrite with x 2 first x = 15 2 x 2 = (sutrcting 9 2 from oth sides) = = 144 x = 144 = 12

10 Locus The locus of n oject is the set of ll the possile positions tht this oject cn occupy. In prticulr the pth of n oject, moving ccording to some rule, is the locus of the oject. (The plurl of locus is loci) sometimes the locus cn e descried in words; sometimes it cn e etter descried y sketch. To find the locus of moving point, lwys sketch few possile positions of the point. 1. The locus of point which is constnt distnce from fixed point is circle locus 2. The locus of point which is constnt locus distnce from fixed line is pir of prllel lines locus 3. The locus of point which is equidistnt from two fixed points is the perpendiculr isector of the line joining the fixed points 4. The locus of point which is equidistnt from two intersecting lines is the pir of lines which isect the ngles etween the fixed lines locus locus