Mensuration Archimedes of Samos (287-212 B.C.) studied at Alexandria as a young man. One of the first to apply scientific thinking to everyday problems, he was a practical man of common sense, He gave proois for finding the area, the volume and the centre of gravity of circles, spheres, conics and spirals. By drawing polygons with many sides, he arrived at a value of rbetween 342 and 312. He was killed in the siege of Syracuse at the age of 75. ‘mass, length, area, volume and capacity in practical situations and in terms of larger or smaller units. Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium and compound shapes derived from these Carry out calculations involving the circumference and area of a circle. Solve problems involving the arc length and sector area as fractions of the circumference and area of a Correa en SEN Cog eee RC eRe Ren tor ree oats Carry out calculations involving the surface area and volume of a sphere, pyramid and eats Carry out calculations involving the areas and volumes of compound shapes. ry3.4 Area Rectangle Trapezium . 7 area=1xb area =1(a+b)ht area=4axb =+x(product > “of diagonals) Exercise 1 For questions 1 to 7, find the area of each shape. Decide which information to use: you may not need all of it. sem = 48 SO Mensurationa. am 8. Find the area shaded. 9. Find the area shaded. T be sm cm 4m scm -—— = 10. A rectangle has an area of 117 m? and a width of 9 m, Find its length. 11. A trapezium of area 105 cm? has parallel sides of length 5 em and 9 cm. ‘How far apart are the parallel sides? 12. A kite of area 252 m* has one diagonal of length 9 m. Find the length of the other diagonal. 13. A kite of area 40 m? has one diagonal 2 m longer than the other. Find the lengths of the diagonals. 14. A trapezium of area 140 cm? has parallel sides 10 cm apart and one of these sides is 16 cm long. Find the length of the other parallel side. 15. A floor 5 m by 20 m is covered by square tiles of side 20 cm. ‘How many tiles are needed? 16. On squared paper draw the triangle with vertices at (1, 1), (5, 3), (3, 5). Find the area of the triangle. 17. Draw the quadrilateral with vertices at (1, 1), (6,2), (5,5) (3, 6)- Find the area of the quadrilateral. 18. A square wall is covered with square tiles. ‘There are 85 tiles altogether along the two diagonals. How many tiles are there on the whole wall? 19. On squared paper draw a 7 x 7 square. Divide it up into nine smaller squares. 20. A rectangular field, 400 m long, has an area of 6 hectares. Calculate the perimeter of the field [1 hectare = 10 000m]. In triangle BCD, sinc = a sin€ © area of triangl bxasinC ‘This formula is useful when two sides and the included angle are known, AreaExample Find the area of the triangle shown. R scm Tem Parallelogram —_—_ » ——_ area=bxh area=ba sin @ Exercise 2 In questions 1 to 12 find the area of AABC where AB = c, AC = band BC =a. (Sketch the triangle in each case.) You will need some basic trigonometry (see page 209). 30°. 2. b=11 cm, a=9cm, C=35°. 3.c=12m, b= 12m, A=67.2%. 4.a=5cem,c=6cm, §= 11.8% 6.a=5em,¢=8cm, B= 142°. 14m, A =32°, Be72°. 1. a=7cm,b= 14cm, 65°. 100°. 10cm, B =32". In questions 13 to 20, find the area of each shape. 8m. 13. 14, 15m ia 101m Mensuration15. wen I | ral 17. 18. X 19. Find the area shaded. 20. 19.1 cm 21. Find the area ofa parallelogram ABCD with AB=7 m, AD = 20 mand BAD = 62°. 22. Find the area of a parallelogram ABCD with AD = 7 m, CD = 1 mand BAD = 65°. 23. In the diagram if AE = ~ AB, find the area shaded. Lem 24, The area of an equilateral triangle ABC is 50 cm’. Find AB. 25, The area of a triangle ABC is 64 cm’. Given AB = 11 cm and BC = 15 cm, find ABC, 26. ‘The area of a triangle XYZ is 11m*, Given YZ=7 mand Xx¥Z= 130°, find XY. 27. Find the length of a side of an equilateral triangle of area 10.2 m°. 28. A rhombus has an area of 40 cm’ and adjacent angles of 50° and 130°. Find the length of a side of the rhombus. 29. A regular hexagon is circumscribed by a circle of radius 3 cm with centre O. A cc a) What is angle EOD? b) Find the area of triangle EOD and hence find the area of the hexagon D ABCDER B Area BU30, Hexagonal tiles of side 20 cm are used to tile a room which measures 6.25 m by 4.85 m. Assuming we complete the edges by cutting up tiles, how many tiles are needed? 31. Find the area of a regular pentagon of side 8 cm. 32. ‘The diagram shows a part of the perimeter of a regular polygon with 1 sides ‘The centre of the polygon is at O and O. a) What is the angle AOB in terms of n? 'b) Work out an expression in terms of » for the area of the polygon. OB unit. ©) Find the area of polygons where n= 6, 10, 300, 1000, 10 000. What do you notice? 33. The area of a regular pentagon is 600 cm’. Calculate the length of one side of the pentagon. 3.2 The circle circumference For any circle, the ratio ( J equal to x. diameter “The value of ris usually taken to be 3.14, but this is not an exact value, Through the centuries, mathematicians have been trying to obtain a better value for m. For example, in the third century A.D., the Chinese mathematician Liu Hui obtained the value 3.14159 by considering a regular polygon having 3072 sides! Ludolph van Ceulen (1540-1610) worked even harder to produce a value correct to 35 significant figures, He was so proud of his work that he had this value of engraved on his tombstone. Electronic computers are now able to calculate the value of x to many thousands of figures, but its value is still not exact, ILwas shown in 1761 that zis an irrational number which, like V2 or V3 cannot be expressed exactly as @ fraction. “The first fifteen significant figures of can be remembered from the number of letters in each word of the following sentence. How I need a drink, cherryade of course, after the silly lectures involving Italian kangaroos. ‘There remain a lot of unanswered questions concerning 7, and ‘many mathematicians today ate still working on them. Mensuration‘The following formulae should be memorised. circumference = ad 2nr area = 707? Example Find the circumference and area of a circle of diameter 8 cm. (Take w= 3.142.) Circumference = 503m’ (1d.p) Exercise 3 For each shape find a) the perimeter, b) the area. All lengths are in cm. Use the x button on a calculator or take 7= 3.142. All the arcs are either semi-circles or quarter circles. a) ‘G) *“—» G2 Ce D +6 + | The circleul. ——_— Example 1 A circle has a circumference of 20 m. Find the radius of the circle. Let the radius of the circle by rm. Circumference = 2ar 2nr = 20 20 an r=3.18 ‘The radius of the circle is 3.18 m (3 s.f.). Example 2 ‘A circle has an area of 45 cm*. Find the radius of the circle. Let the radius of the circle by rem. 3.78 (3 sf) ‘The radius of the circle is 3.78 cm. Exercise 4 Use the z button on a calculator and give answers to 3 s.f. 1. A circle has an area of 15 em?, Find its radius. 2. A circle has a circumference of 190m. Find its radius. 3. Find the radius ofa circle of area 22 km’, 4, Find the radius ofa circle of circumference 58.6 cm. 5. A circle has an area of 16 mm*, Find its circumference. 6. A circle has a circumference of 2500 km. Find its area. Mensuration 12. i7. A circle of radius 5 cm is inscribed inside a square as shown. Find the area shaded. 8. A circular pond of radius 6m is surrounded by a path of width 1m. sem a) Find the area of the path. ) ‘The path is resurfaced with Astroturf which is bought in packs each containing enough to cover an area of 7m’. How many packs are required? 9. Discs of radius 4cm are cut from a rectangular plastic sheet of length 84cm and width 24¢m. a) How many complete dises can be cut out? Find: ) the total area of the discs cut ©) the area of the sheet wasted. 10. The tyre ofa car wheel has an outer diameter of 30cm. How many times will the wheel rotate on a journey of 5km? 11. A golf ball of diameter 1.68 inches rolls a distance of 4 m in a straight line. How many times does the ball rotate completely? (1 inch = 2.54cm) 12. 100 yards of cotton is wound without stretching onto a reel of diameter 3 em. How many times does the reel rotate? (1 yard = 0.914m. Ignore the thickness of the cotton.) 13. A rectangular metal plate has a length of 65cm and a width of 35cm. It is melted down and recast into circular discs of the same thickness. ‘How many complete discs can be formed if a) the radius of each disc is 3cm b) the radius of each disc is Locm? 14, Calculate the radius of a circle whose area is equal to the sum of the areas, of three circles of radii 2cm, 3cm and 4.cm respectively. 15. The diameter of a circle is given as 10cm, correct to the nearest cm. Calculate: a) the maximum possible circumference b) the minimum possible area of the circle consistent with this data, 16. A square is inscribed in a circle of radius 7 em. Find: a) the area of the square b) the area shaded. 17. An archery target has three concentric regions. The diameters of the regions are in the ratio 1:2:3. Find the ratio of their areas. 18. The farmer has 100 m of wire fencing. What area can he enclose if he makes a circular pen? The circle19. The semi-circle and the isosceles triangle have the same base AB and the same area. Find the angle x. 20. Lakmini decided to measure the circumference of the Earth using a very long tape measure. She held the tape measure Lm from the surface of the (perfectly spherical) Earth all the way round. When she had finished her friend said that her measurement gave too large an answer and suggested taking off 6 m, Was her friend correct? [Take the radius of the Earth to be 6400 km (if you need it).] 21. ‘The large circle has a radius of 10 cm. Find the radius of the largest circle which will fit in the middle. <— 3-3 Arc length and sector area wer e Arc length, ! = —— x 23 rc leng! 5q0 27" We take a fraction of the whole circumference depending on the angle at the centre of the circle. secio“A An ixnet Sector area, A= 355 ‘We take a fraction of the whole area depending on the angle at the centre of the circle, STEM MensurationExample 1 Find the length of an arc which subtends an angle of 140° at the centre of a circle of radius 12 cm. Arc length = 350 2% x x12 8 =n 3 =293cm(1dp.) Example 2 ‘A sector of a circle of radius 10 cm has an area of 25 cm*, Find the angle at the centre of the circle. Let the angle at the centre of the circle be 8. aa fx axio? =25 360 2X100 @ = 28.6° (3 sf.) ‘The angle at the centre of the circle is 28.6°. Exercise 5 [Use the button on a calculator unless told otherwise.] 1. Are AB subtends an angle @ at the centre of circle radius r. Find the arc length and sector area when: a) r=4cm,0=30° b) r=10cm,0=45° ©) r=2em,0=235%, In questions 2 and 3 find the total area of the shape. OA =2.cm, OB=3 cm, OC=5 cm, OD=3cm. x % ‘Arc length and sector area rey3. K ON =6cm,OM=3«m, OL = 2m, OK =6 cm. 4, Find the shaded areas. 5. In the diagram the arc length is land the sector area is A. a) Find 8, when r=5 em and 1=7.5 em, b) Find @, when r= ©) Find r, when @ mand A=2m*, 5° and | cm. 6. The length of the minor are AB of a circle, centre O, is em and the length of the major arc is 227 cm. Find a) the radius of the circle b) the acute angle AOB, 7. The lengths of the minor and major arcs of a circle are 5.2 em and 19.8 respectively. Find: minor ae a) the radius of the circle b) the angle subtended at the centre by the minor arc. 8. A wheel of radius 10 cm is turning at a rate of 5 revolutions per minute. Calculate: a) the angle through which the wheel turns in 1 second b) the distance moved by a point on the rim in 2 seconds. 9. The length of an arc of a circle is 12 cm. The corresponding sector area is 108 cm’, Find: a) the radius of the circle b) the angle subtended at the centre of the circle by the arc. Mensuration10. the length of an arc of a circle is 7.5 em. The corresponding sector area is 37.5 cm*, Find: a) the radius of the circle b) the angle subtended at the centre of the circle by the are. 11. In the diagram the arc length is land the sector area is A. a) Find J, when 0= 72° and A=15 em’. b) Find J, when @= 135° and A = 162 cm*, ©) Find A, when I= 1] emand r=5.2. em. 12, A long time ago Dulani found an island shaped like a triangle with three straight shores of length 3 km, 4 km and 5 km. He said nobody could come within 1 km of his shore. What was the area of his exclusion zone? 3-4 Chord of acircle La ‘The line AB isa chord. ‘The area of a circle cut off by a chord is called a segment. In the diagram the minor segment is shaded and the major segment is unshaced. a) ‘The line from the centre of a circle to the midpoint of a chord bisects the chord at right angles. b) ‘The line from the centre of a circle to the midpoint of a chord bisects the angle subtended by the chord at the centre of the circle. Example XY isa chord of length 12 cm ofa circle of radius 10 cm, centre O. Calculate: a) the angle XOY b) the area of the minor segment cut off by the chord XY. Chord ofa circlea) Let the midpoint of XY be M. & MY =6cm b) Area of minor segment = area of sector XOY — area of AXOY area of sector xO = 274 x9 x10? 360 = 64.32 cm’. area of AXOY = +x 10% 10 sin 73.74° = 48.00 cm? Area of minor segment = 64.32 — 48.00 = 16.3 cm* (3 sf.) Exercise 6 Use the button on a calculator. You will need basic trigonometry (page 209). 1, The chord AB subtends an angle of 130° at the centre O. ‘The radius of the circle is 8 em. Find: a) the length of AB a b) the area of sector OAB c) the area of triangle OAB NTS - 4) the area of the minor segment (shown shaded). 2. Find the shaded area when: a) r=6cm, @=70° b) r= 14cm, 0= 104° Zs ©) r=Sem,0=80° — 3, Find @and hence the shaded area when: a a) AB=10cm,r= 10cm (> b) AB=8em,r=5cm 4, How faris a chord oflength 8 cm from the centre of a circle of radius 5 em? ® 5. How far isa chord of length 9 cm from the centre of a circle of radius 6 cm? Mensuration6. The diagram shows the cross-section of a cylindrical pipe with water lying in the bottom. a) Ifthe maximum depth of the water is 2 cm and the radius of the pipe is 7 em, find the area shaded. b) What is the volume of water in a pipe length of 30 cm? 7. An equilateral triangle is inscribed in a circle of radius 10 cm. Find: a) the area of the triangle b) the area shaded. 8. An equilateral triangle is inscribed in a circle of radius 18.8 cm. Find: a) the area of the triangle b) the area of the three segments surrounding the triangle. 9. A regular hexagon is circumscribed by a circle of radius 6 cm. Find the area shaded, 10. A regular octagon is circumscribed by a circle of radius r cm. Find the area enclosed between the circle and the octagon. (Give the answer in terms of r.) 11. Find the radius of the circle: a) when 0=90°, A= 20cm? ‘b) when @= 30°, A ©) when @= 150°, A= 114 cm* 5 cm? 12, The diagram shows a regular pentagon of side 10 em with a star inside. Calculate the area of the stat. 4d KK 10m Chord ofa circletr} 3-5 Volume Prism A prism is an object with the same cross-section throughout its length. Volume of prism = (area of cross-section) length =Ax1 A cuboid is a prism whose six faces are all rectangles. A cube is a special case of a cuboid in which all six faces are squares. Cylinder radius =r height = A cylinder is a prism whose cross-section is a circle. Volume of cylinder = (area of cross-section) x length. Volume = rr*h Example Let the height of the cylinder be ht cm. arh = 500 3.1428" x= 500 500 3.142 64 .49 (3 sf.) ‘The height of the cylinder is 2.49 em. h Calculate the height ofa cylinder of volume 500 cm* and base radius 8 em. Exercise 7 1. Calculate the volume of the prisms. All lengths are in cm. a) EI b) SNE 4) oS) O Mensuration " E<> f) —s— oe2. Calculate the volume of the following cylinders: a) r=4cm, h=10cm b) r=llm,— h=2m OQ r=2lem, — h=0.9em 3. Find the height ofa cylinder of volume 20cm and radivs4em. Remember itte= 1000 cms, 4, Find the length of a cylinder of volume 2 litres and radius 10cm. 5. Find the radius ofa cylinder of volume 45 cm' and length 4 em, 6. A prism has volume 100 cm! and length 8 cm. If the cross-section isan equilateral triangle, find the length of a side of the triangle. 7. When 3 litres of oil are removed from an upright cylindrical can, the level falls by 10 cm. Find the radius of the can. 8. A solid cylinder of radius 4 cm and length 8 cm is melted down and recast into a solid cube. Find the side of the cube. ). A solid rectangular block of copper 5 em by 4.em by 2cm is drawn out to make a cylindrical wire of diameter 2 mm. Calculate the length of the wire. 10. Water flows through a circular pipe of internal diameter 3 cm at a speed of 10 cm/s. Ifthe pipe is full, how much water flows from the pipe in one minute? (Answer in litres.) LL. Water flows from a hose-pipe of internal diameter 1 cm at a rate of S litres per minute. At what speed is the water flowing through the pipe? 12. A cylindrical metal pipe has external diameter of 6 cm and internal diameter of 4 em, Calculate the volume of metal in a pipe oflength Im. If 1 em* of the metal has a mass of 8g, find the mass of the pipe. 13. For two cylinders A and B, the ratio of lengths is 3: 1 and the ratio of diameters is 1:2. Calculate the ratio of their volumes. 14, A machine makes boxes which are either perfect cylinders of diameter and length 4 cm, or perfect cubes of side 5 cm. Which boxes have the greater volume, and by how much? (Take = 3) 15. Natalia decided to build a garage and began by calculating the number of bricks required. The garage was to be 6m by 4m and 2.5 m in height. Each brick measures 22 cm by 10 cm by 7 cm. ‘Natalia estimated that she would need about 40 000 bricks. Is this a reasonable estimate? Volume16. A cylindrical can of internal radius 20 cm stands upright on a flat surface. It contains water to a depth of 20 cm. Calculate the rise in the level of the water when a brick of volume 1500 cm’ is, immersed in the water. 17. A cylindrical tin of height 15 em and radius 4 cm is filled with sand from a rectangular box. How many times can the tin be filled if the dimensions of the box are 50 cm by 40 em by 20 cm? 18, Rain which falls onto a flat rectangular surface of length 6m. and width 4m is collected in a cylinder of internal radius 20 cm. What is the depth of water in the cylinder after a storm in which Lem of rain fell? Pyramid Volume = ! (base area) x height. S| | a za Figure (i) Figure (i) shows a cube of side 2a broken dowa into six pyramids of height a as shown in Figure (ii). Ifthe volume of each pyramid is V; then 6V = 2a x 2ax2a V=tx (2a)' x20 so Vet (2a) xa V=1 (base area) x height EI MensurationCone Volume = + 17h (note the similarity with the pyramid) Sphere 4p Volume = * nr Example 1 A pyramid has a square base of side 5m and vertical height 4m. Find its volume. Volume of pyramid = +(5x5)x4 =331m? 3 Example 2 Calculate the radius of a sphere of volume 500 cm’. Let the radius of the sphere be r em. As = ar’ = 500 3 » _ 3x50 re an (2 is =) = 492(3s£) 7 ‘The radius of the sphere is 4.92 em. Exercise 8 Find the volumes of the following objects: 1. cone: height = 5 em, radius = 2m 2. sphere: radius = 5 em Volume3, sphere: radius = 10 em 4. cone: height = 6 cm, radius = 4 em 5. sphere: diameter = 8 cm 6. cone: height = x cm, radius = 2x em 7. sphere: radius = 0.1m 1 8. cone: height em, radius = 3 cm a 9. pyramid: rectangular base 7 em by 8 em; height = 5 cm 10. pyramid: square base of side 4 m, height = 9m LL. pyramid: equilateral triangular base of side = 8 cm, height = 10 em 12. Find the volume of a hemisphere of radius 5 em. 13. A cone is attached to a hemisphere of radius 4 cm. If the total height of the object is 10 cm, find its volume. 14, A toy consists of a cylinder of diameter 6 em ‘sandwiched’ between a hemisphere and a cone of the same diameter. If the cone is of height 8 cm and the cylinder is of height 10 cm, find the total volume of the toy. 15. Find the height of a pyramid of volume 20m and base area 12m?, 16. Find the radius of a sphere of volume 60 cm* 17, Find the height of a cone of volume 2.5 litre and radius 10 cm. 18. Six square-based pyramids fit exactly onto the six faces of a cube of side 4 cm. If the volume of the object formed is 256 cm‘, find the height of each of the pyramids. 19. A solid metal cube of side 6 cm is recast into a solid sphere. Find the radius of the sphere. 20. a hollow spherical vessel has internal and external radii of 6 em and 6.4 cm respectively. Calculate the mass of the vessel if it is made of metal of density 10 g/cm’. 21. Water is flowing into an inverted cone, of diameter and height 30 cm, ata rate of litres per minute, How long, in seconds, will it take to fill the cone? 22. A solid metal sphere is recast into many smaller spheres. Calculate the number of the smaller spheres if the initial and final radii are as follows: SEEM Mensuration23, 24. 25. 26. 27. 28. a) initial radius = 10 cm, final radius= 2. cm ) initial radius =7 cm, final radius = + cm ©) initial radius = 1 m, final radius = + cm. ‘A spherical ball is immersed in water contained in a vertical cylinder. Assuming the water covers the ball, calculate the rise in the water level if: a) sphere radius =3 cm, cylinder radius = 10 em b) sphere radius = 2.cm, cylinder radius = 5 em. A spherical ball is immersed in water contained in a vertical cylinder. The rise in water level is measured in order to calculate the radius of the spherical ball. Calculate the radius of the ball in the following cases: a) cylinder of radius 10 em, water level rises 4 em ‘b) cylinder of radius 100 cm, water level rises 8 cm. One corner of a solid cube of side 8 em is removed by cutting. through the midpoints of three adjacent sides. Calculate the volume of the piece removed. ‘the cylindrical end of a pencil is sharpened to produce a perfect cone at the end with no overall loss of length. Ii the diameter of the pencil is 1 em, and the cone is of length 2 cm, calculate the volume of the shavings. Metal spheres of radius 2 cm are packed into a rectangular box of internal dimensions 16 cm x 8 cm x 8 cm. When 16 spheres are packed the box is filled with a preservative liquid. Find the ‘volume of this liquid. ‘The diagram shows the cross-section of an inverted cone of height MC = 12 cm. If AB = 6 cm and XY = 2 cm, use similar triangles to find the length NC. (You can find out about similar triangles on page 147.) Volume [I29. An inverted cone of height 10 cm and base radius 6.4 em contains water to a depth of 5 cm, measured from the vertex. Calculate the volume of water in the cone. 30. An inverted cone of height 15 cm and base radius 4 cm_ contains water to a depth of 10 cm. Calculate the volume of water in the cone. 31. An inverted cone of height 12 cm and base radius 6 cm contains 20 cm* of water. Calculate the depth of water in the cone, measured from the vertex. 32. A frustum is a cone with ‘the end chopped off A bucket in the 19cm > shape of a frustum as shown has diameters of 10 cm and 4 em at its ends and a depth of 3 cm. Calculate the volume of the bucket. ! dem ion 33, Find the volume of a frustum with end diameters of 60 cm and 20cm and a depth of 40 cm. 34, The diagram shows a sector of a circle of radius 10 cm. a) Find, as a multiple of 7, the arc length of the sector. The straight edges are brought together to make a cone. # Calculate: 'b) the radius of the base of the cone ©) the vertical height of the cone. 35. Calculate the volume of a regular octahedron whose edges aze all 10cm. 36. A sphere passes through the eight corners of a cube of side 10 cm. Find the volume of the sphere. 37. ind the volume of a regular tetrahedron of side 20 cm. 38. Find the volume of a regular tetrahedron of side 35 em. SEZ Mensuration3-6 Surface area We are concerned here with the surface areas of the curved parts of cylinders, spheres and cones. ‘Ihe areas of the plane faces are easier to find, a) Cylinder b) Sphere ) Cone ‘Curved surface area = 2mrh Surface area = 4° Curved surface area = ar! where Lis the slant height. Example 1 Find the ‘ofa! surface area of a solid cone of radius 4 em and vertical height 3 cm. Let the slant height of the cone be / cm. P=34+4° — (Pythagoras’ theorem ) 1=5 = Curved surface area = 1X 4x5. = 20m cm’ Area of end face = 1x4? = 167 om? ‘Total surface area = 207 +16 36m cm? =113cm? (3s) Example 2 Find the surface area of a prism, whose length is 50 em and whose cross-section is a regular hexagon with side length 10 cm and area 172 cm’, There is no general formula for the surface area of a prism. Area of the two hexagons = 2x 172 =344cm* ‘Area of the six rectangular faces = 6 x 50 x 10 = 3000 cm? ‘Total area “idem Surface areaExercise 9 ‘Use the z button on a calculator unless otherwise instructed, 1. Copy the table and find the quantities marked”. (Leave win your answers.) solid object | radius _| vertical height | curved surface area | total surface area a) | sphere 3cm * b) | cylinder 4cm Scm . ©) | cone 6cm 8cm > 4) | cylinder 07m im * ¢) | sphere 10m * f) | cone Som lem > 8) (cylinder 6mm _|_10mm » h) | cone 21cm 44cm * i) | sphere 0.01 m + j) [hemisphere 7em * » 2. Find the radius of a sphere of surface area 34cm’, 3. Find the slant height of a cone of. ‘curved surface area 20 cm* and radius 3 cm. 4. Find the height of a solid cylinder of radius 1 em and fotal surface area 28 em’. 5. Copy the table and find the quantities marked”. (Take m= 3) object radius | vertical height | curved surface area | total surface area a) [cylinder dem * Tem b) | sphere * 192 cm? ©) [cone cm * 60cm? d) | sphere * 0.48 m? e) [cylinder Sem > 330 cm* £) [cone 6cm * 225 cm? 2) (cylinder 2m * 108 m? 6. A solid wooden cylinder of height 8 cm and radius 3 cm is cut in two along a vertical axis of symmetry. Calculate the total surface area of the two pieces. 7. A tin of paint covers a surface area of 60m? and costs $4.50. Find the cost of painting the outside surface of a hemispherical dome of radius 50 m. (Just the curved part.) 8. A solid cylinder of height 10 cm and radius 4 cm is to be plated », Find the cost of the plating. with material costing $11 per cr 9, Find the volume of a sphere of surface area 100 cm*, 10. Find the surface area of a sphere of volume 28 cm’. 11. Calculate the total surface area of the combined cone/cylinder/hemisphere. SE Mensuration 120m oom __, $e12. A man wants to spray the entire surface of the Earth (including the oceans) with a new weed killer. Ifit takes him 10 seconds to spray 1 m’, how long will it take to spray the whole world? (Radius of the Earth = 6370 km ignore leap years) 13. An inverted cone of vertical height 12 cm and base radius 9 cm contains water to a depth of 4 cm, Find the area of the interior surface of the cone not in contact with the water. 14. A circular piece of paper of radius 20 cm is cut in half and each half is made into a hollow cone by joining the straight edges, Find the slant height and base radius of each cone. 15. A golf ball has a diameter of 4.L cm and the surface has 150 dimples of radius 2 mm. Calculate the total surface area which is exposed to the surroundings. (Assume the ‘dimples’ are hemispherical.) 16. A cone of radius 3 cm and slant height 6 cm is cut into four identical pieces. Calculate the total surface area of the four pieces. dimple of radius 2mm — In questions 17 to 20 find the surface area of each prism. = 7. em 18. 2em Sem Tocm —sm— 20. 10cm, som Surface arearr Example Alem by Lem square measures 10 mm by 10 mm, ‘The area of the square in mm is therefore 10x 10 = 100mm’, ‘There are similar area conversions for m* into em* and km* into m*: 1m? = 100x 100 = 10000cm? 1km?= 1000 x 1000 = 1000000 m? A Lem by Lem by 1 cm cube measures 10mm by 10 mm by 10 mm. ‘The volume of the cube is therefore 10 x 10 x 10= 100mm’. Likewise, | m*= 100 x 100 x 100 = 1 000000cm* Lem =10 mm Lem=l0mm Tem =10 mm fem = 10 mm Lem = 10mm Exercise 10 Copy and complete. 1. 2m’ 3.1600mm?= cm’ 4. 48mm? 6. 26m cm? 7. 8600 cm’ 9, 5kan mm 10. 4500 000 12. 21cm mm 13. 48000mm*= — cm’ 14.6m°= cm 15. 28000000cm?= —m* 16. A cuboid measures 3cm by 2cm by 4cm. a) Find the volume in mm*, 17. A rectangle measures 40cm by 80cm. Find the area in m2, 18. A sphere has radius 6.2m. a) Find the surface area in cm’, —_b) Find the volume in mv’ 19. A square-based pyramid has base area 300 cm? and height 40 mm. Find the volume in mm*, 20. A cylinder has volume 1200 cm’, The length of the cylinder is 42 mm, Find the radius of the cylinder in mm, Revision exercise 3A 1. Find the area of the following shapes: Qo. A a Tex, = jocm b) Re d) 3 : em . i sem — rie Mensuration. b) Find the surface area in mm. ©) Find the volume in em’. t 4em 12.a) A circle has radius 9m. Find its circumference and area. b) A circle has circumference 34 em. Find its diameter. ©) Accircle has area 50 cm’, Find its radius. 3. A target consists of concentric circles of radii 3.cmand 9m. a) Find the area of A, in terms of area of B b) Find the ratio = area of A 4, In Bigure 1 a circle of radius 4 cm is inscribed in a square, In Figure 2 a square is inscribed in a circle of radius 4 em. Calculate the shaded area in each diagram. Figure 1 \ ~~ Figure 2 5. Given that OA = 10 cm and AOB = 70° (where O is the centre of the circle), calculate: a) the arclength AB a b) the area of minor sector AOB. al je "The shape above is made by removing a small semi-circle from a large semi-circle, AM = MB= 12cm BI Calculate the area of the shape. Cambridge IGCSE Mathematics 0580 Paper 2 Q10 November 2007 5. Nottoscale A n —Tean ‘The largest possible circle is drawn inside a semi-circle, as shown in the diagram. ‘The distance AB is 12 centimetres. (a) Find the shaded area. la] (b) Find the perimeter of the shaded area. fa) Cambridge IGCSE Mathematics 0580 Paper 2 Q23 June 2007 6. Nottoscale The diagram shows part of a logo that has been designed for an engineering company. OAD and OFG are sectors, centre O, with radius 12 em and angle 50°. B,C, Band H lie on a circle, centre O, and radius 4em. Calculate, correct to 3 significant figures, the area shaded. 4] SEPM Mensuration2 Notto scale 4 2 D f A, B, Cand D lie on a circle, centre O, radius 8 em. AB and CD are tangents to a circle, centre O, radius 4 cm. ABCD is a rectangle. (a) Calculate the distance AE 2 (b) Calculate the shaded area, (3) Cambridge IGCSE Mathematics 0580 Paper 2 Q21 November 2005 ———— Not toscale A solid metal bar is in the shape of a cuboid of length of 250 cm. ‘The cross-section is a square of side x em. ‘The volume of the cuboid is 4840 em’. (a) Show that x= 4.4, 2) (b) ‘The mass of 1 cm’ of the metal is 8.8 grams. Calculate the mass of the whole metal bar in kilograms. (21 (©) A box, in the shape of a cuboid measures 250 cm by 88 em by fem. 120 of the metal bars fit exactly in the box. Calculate the value of h. (21 (a) One metal bar, of volume 4840 cm’, is melted down to make 4200 identical small spheres. All the metal is used. i) Calculate the radius of each sphere, Show that your answer rounds to 0.65 cm, correct to 2 decimal places. (4) [The volume, V, of a sphere, radius r, is given by V=42r°.] Examination-style exercise 3Bii) Calculate the surface area of each sphere, using 0.65 cm for the radius, i [The surface area, A, of a sphere, radius r, is given by A = 4ar*.| Calculate the total surface area of all 4200 spheres as a percentage of the surface area of the metal bar. [4] Cambridge IGCSE Mathematics 0580 Paper 4 Q7 June 2009 9. Nottoscale A circle, centre O, touches all the sides of the regular octagon, ABCDEFGH shaded in the diagram. ‘The sides of the octagon are of length 12 cm. BA and GH are extended to meet at P. HG and EF are extended to meet at Q (a) i) Show that angle BAH is 135°. [2] ii) Show that angle APH is 90°. it} (b) Calculate i) the length of PH, [21 ii) the length of PQ, ie) iii) the area of triangle APH, [2] iv) the area of the octagon. [3] (©) Calculate i) the radius of the circle, 2 ii) the area of the circle as a percentage of the area of the octagon. B31 Cambridge IGCSE Mathematics 0580, Paper 4 Q5 June 2008 SEE Mensuration10. Paul and Debbie buy their son a magic set. ‘The box is a prism with a cross-section that is a trapezium, as shown in the diagram. sen 50cm Dan a) Calculate the area of the cross-section, 2] b) Calculate the volume of the box. (21 ©) Calculate the surface area of the box. (3) Examination-style exercise 3B