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IGCSE Higher Sheet H7-1 4-08d-1 3D Pythagoras Sheet H7-2 4-08d-2 3D Pythagoras Sheet H7-3 4-08d-3 3D Pythagoras Sheet H7-4 4-08e-1 3D Trigonometry Sheet H7-5 4-08e-2 3D Trigonometry Sheet H7-6 4-08e-3 3D Trigonometry Sheet H7-7 4-08e-4 3D Trigonometry Sheet H7-8 4-09-1 Sectors Sheet H7-9 4-09-2 Sectors Sheet H7-10 4-09-3 Area and Volume - Non-Calc Sheet H7-11 4-09-4 Area and Volume Sheet H7-12 4-09-5 Area and Volume Sheet H7-13 4-09-6 Area and Volume Sheet H7-14 4-09-7 Area and Volume Sheet H7-15 4-10-1 Similar Shapes Sheet H7-16 4-10-2 Similar Shapes Sheet H7-17 4-10-3 Similar Shapes Sheet H7-18 4-10-4 Similar Shapes Sheet H7-19 4-10-5 Similar Shapes Sheet H7-20 5-01a-1 Vectors Sheet H7-21 5-01a-2 Vectors Sheet H7-22 5-01a-3 Vectors Sheet H7-23 5-01a-4 Vectors



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Sheet H7-1 4-08d-1 3D Pythagoras 1. ABCDEFGH is a cuboid with AB = 12cm, AD =3cm and AE =4cm.

Find the following lengths: (a) AC (to 3sf) (b) BE (to 3sf) (c) ED (d) FD

2.

In the pyramid ABCDE, the rectangular base ABCD is horizontal and EM is vertical. M is the midpoint of AC. AB = 6cm, AD = 8cm and AE = 13cm. Find the following: (a) AM. (b) the height of the pyramid.

3. In the wedge show below the rectangle BEFC is perpendicular to rectangle ABCD. AB = 6cm, BC = 9cm and BE = 4cm.

(a) Find the following: (i) AC (to 3sf) (ii) BF (to 3sf)

(b) Find the length of the line which joins AD to EF in such a way that the line makes the smallest possible angle with the base ABCD.

4. In the pyramid ABCDE, the rectangular base

ABCD is horizontal and EM is vertical. M is the midpoint of AC. AB = 10m, AD = 12m and

EM = 18m. Find the following: (a) AM (to 3sf) (b) AE (to 3sf) (c) EN (to 3sf) where N is the midpoint

of AB.

CDE

F

A B



Sheet H7-2 4-08d-2 3D Pythagoras

1. A man walks 150m due East of a tower and then 360m due North. (a) How far is he from the base of the tower. (b) If the tower is 100m tall then how far is it (to 3sf) from the top of the tower to the man

(ignoring the height of the man).

2. A tin in the shape of cuboid is 6cm wide, 8cm long and 5cm tall. (a) What is the length of a diagonal of the base of the tin? (b) What is the longest thin pole (to 3sf) that can be fitted in the tin (ignoring the

thickness of the pole)?

3. A man walks 30m due West from the foot of an aerial mast which is 40m high. He then walks 50m North. How far (to 3sf) is he from the top of the mast?

4. One plane is 7km due South of an airport and another plane is 5km due East of the same

airport. If the difference in their heights is 500m then find the distance between the two planes (to 3sf).

5. A man in a car is 12km South West of a certain town. A plane is 5km North West of the

same town. At what height (to the nearest 100m) is the plane flying if the distance between the man and the plane is 13.1km?

6. A long, thin metal box is 5mm, 12mm high and 84mm long. What is the longest rod that

can be fitted into this box? 7. A box is such that it is ratio of its length to its width is 4:3. It is 91cm high and the largest

pole which can be fitted in the box is 109cm. Find the width of the box. (NB Let the width of the box be 3x).

8. A cuboid has lengths in the ratio 3:4:12. If the distance from one corner of the box to the

corner which is furthest away is 65mm then find the dimensions of the box. 9. An office block is 200m further away from an underground station than a tall monument is

from the same station. The office block is due East of the station and the monument is due North of the station.

The distance between the top of the office block and the top of the monument is 600m and the office block is 100m taller than the monument.

(a) If x is the distance from the foot of the monument to the underground station then write down an equation involving x. (b) Show that 2 200 155000 0x x+ − = (c) Solve this equation to find x (to 3sf).



Sheet H7-3 4-08d-3 3D Pythagoras 1. A plane is 2km due South of an airport and is flying at a height of 1200m. A helicopter is flying at a height of 400m and is 1km due West of the same airport. Find the distance between the plane and the helicopter (to 3sf). 2. A cable is attached to the top of a vertical pole which is 120m high. The other end of the

cable is attached to a point of the ground which is 18m to the North of the foot of the pole and 24m to the East of the foot of the pole. Find the length of the cable (to 3sf).

3. The sides of a cuboid are in the ratio 3:4:5. If the greatest distance between two corners of

the cuboid is 125cm then find the length of the shortest side (to 3sf). 4. A cuboid is such that its height is 1m longer than its width and its length is 2m longer than

its height. (a) If w is the width of the cuboid then find expressions for the height and the length in terms of w.

If the longest diagonal in the cube is 6m then (b) Write down an equation involving w. (c) Solve this to find the width of the cuboid (to 3sf). 5. In a three dimensional co-ordinate system the point P has co-ordinates (x, y, z) where x, y

and z are three consecutive positive integers (x being the smallest). If the distance of the point P from the origin is at least 175 then find the minimum possible value of x.



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Sheet H7-4 4-08e-1 3D Trigonometry

1. In the pyramid ABCDE, the square base ABCD is horizontal and EM is vertical. M is the midpoint of AC and N is the midpoint of AB. AB = 15cm and EM = 11cm. (a) Find AM (to 3sf). (b) Show that 3.15AE = cm (to 3sf). (c) Calculate the angle (to 1dp) between

the planes AEB and ABCD. (d) Calculate the angle (to 1dp)

between the line AE and the plane ABCD.

2. In the pyramid ABCDE, the rectangular base ABCD is horizontal and EM is vertical. M is

the midpoint of AC and N is the midpoint of BC. AB = 15cm, BC=20cm and EM = 12cm. (a) Show that cm 5.12CM = (b) Calculate the angle (to 1dp)

between the line EC and the horizontal base.

(c) Calculate the angle (to 1dp) between the planes BCE and the horizontal base.

(d) Calculate the angle (to 1dp) between the planes ADE and BCE.

PTO



Sheet H7-4 4-08e-1 3D Trigonometry (cont.)

3. In the pyramid ABCDE, the square base ABCD is horizontal and EM is vertical. M is the midpoint of AC and N is the midpoint of AB. AB = 20cm and EM = 25cm. (a) Show that cm 210BM = (b) Calculate the angle (to 1dp) between the line EB and the plane ABCD. (c) Find EB (to 3sf). (d) Calculate the angle (to 1dp) between the planes AEB and ABCD. (e) Calculate the angle ABE (to 1dp). (f) Calculate the length of AX (to 3sf)

where X is the point on EB such that AX is perpendicular to EB.

(g) Name the angle between the planes AEB and EBC.

(h) Calculate the angle (to 1dp) between the planes AEB and EBC.




Abccba cos2222 −+= and bc

acbA2

cos222 −+

=

1. ABCDEFGH is a cuboid with AB = 25cm, AD =13cm and AE =18cm.

(a) Which of the following angles are right angles ABC ABG AFH ADG HFC DGF? (b) Find the following lengths (leave as square

roots) (i) AC (ii) EB (iii) BG (iv) DF

(c) Show that the angle GAC is 32.6 (to 1dp). (d) Calculate the angle BEC (to 1dp) using trigonometry. (e) Calculate EBG (to 1dp) using the Cosine rule.

2.

In the pyramid ABCDE, the rectangular base ABCD is horizontal and EM is vertical. M is the midpoint of AC. AB = 35cm, AD = 20cm and AE = 24cm. (a) Find the following lengths (to 3sf)

(i) AM (ii) EM

(b) Find the following angles (to 1dp) (i) EAM (ii) AEB (iii) DEB (i.e. DEM2 ∠× )

PTO

25cm

13cm

18cm

35cm

20cm 24cm



Sheet H7-5 4-08e-2 3D Trigonometry (cont.)

3. The diagram below represents a gift box. ABCD and EFGH are horizontal squares of side lengths 12cm and 20cm respectively. The 4 slanting sides AE, BF, CG and DH all are 28cm long and all make the same angle with the horizontal base. Let X be the point on EF such that AX is perpendicular to EF. (a) Explain why the length of EX is 4cm.

(b) Calculate the angle AEF (to 1dp). (c) If Y is the point on the top of the box which is vertically above B then find FY (to 3sf). (HINT: Consider a bird’s eye view of EFGH) (d) Hence find the vertical height of the gift box (to 3sf). (HINT: Draw BYF)

28c

20c




1. ABCDEFGH is a cuboid with AB = 20mm, AD = 12mm and AE =15mm. (a) Show that the angle between the planes

ABGH and ABCD is °3.51 (to 1dp). (b) Find the angle (to 1dp) between the

planes EHCB and ADHE. (c) Find the angle (to 1dp) between the line

AG and the plane DCGH. (d) Find the angle (to 1dp) between the line

BG and the plane DCGH.

2. In the pyramid ABCDE, the square base ABCD is horizontal and EM is vertical. M is the midpoint of AC and N is the midpoint of AB. AB = 120cm and EM = 180cm. (e) Find BM (to 3sf). (f) Show that 1160EB = cm. (g) Calculate the angle (to 1dp) between

the planes AEB and ABCD. (h) Calculate the angle (to 1dp) between

the line EB and the plane ABCD. (i) Calculate the angle ABE (to 1dp). (j) Calculate the length of AX (to 3sf)

where X is the point on EB such that AX is perpendicular to EB.

3. The diagram below represents the roof of a house.

ABCD is horizontal with AB = 10m and BC = 26m. EF is horizontal, is 20m long and is 2m above the horizontal plane ABCD. AEB and DFC make the same angle with ABCD. BCFE and ADFE make the same angle with ABCD. N is in the plane ABCD and is vertically below E. M is the midpoint of AB. X is the point on BC such that NXC is a right angle.

(a) Write down the lengths of EN and MN. (b) Find the angle (to 1dp) between the planes AEB and ABCD. (c) Calculate the length NC (d) Find the angle (to 1dp) which the line EC makes with the plane ABCD. (e) Find the angle (to 1dp) between the planes EBCF and ABCD. (f) Find the angle (to 1dp) between the planes EBCF and ADFE.




1. ABCDEFGH is a cuboid with AB = 30cm, AD =10cm and AE =20cm.

Calculate the following: (a) The length of EB (to 3sf) (b) The length of HB (to 3sf) (c) The angle which EB makes with the base

of the cuboid (to 1dp) (d) The angle which EHBC makes with

BCFG (to 1dp) (e) The angle which ABGH makes with

HGCD (to 1dp). 2.

In the pyramid ABCDE, the rectangular base ABCD is horizontal and EM is vertical. M is the midpoint of AC. AB = 15cm, AD = 18cm and EM = 12cm. Find the following: (a) the length of AM (to 3sf) (b) the length of AE (to 3sf) (c) the angle which EBC makes with ABCD (to

1dp) (d) the angle which EB makes with ABCD (to

1dp) 3. The diagram opposite represents a tent. The base ABCD is

a horizontal rectangle with AB = 1.4m and AD = 2.3m. EF is horizontal and EF = 1.7m. The height of the tent is 1m. The ends AEB and DFC make the same angle with the horizontal. The sides EBFC and ADFE make the same angle with the horizontal. (a) Calculate the length of EM where M is the midpoint of AB ( to 3sf) (b) Show that the length of EB is 1.26m (to 3sf) (c) Calculate the angle which AEB makes with the horizontal base (to 1dp) (d) Calculate the angle which EBFC makes with the horizontal base (to 1dp) (e) Show that the angle EBC is °2.76 (to 1dp) (f) Calculate the angle EBA (to 1dp) (g) Calculate the length of BX where X is the point on EB such that MX is perpendicular

to EB (to 3sf) (h) Use (e) and (g) to find the distance BY (to 3sf) where Y lies on the line BC and XY is

perpendicular to EB. (i) Name and find the angle between AEB and EBCF (to 1dp).

Abccba cos2222 −+= and bc

acbA2

cos222 −+

=



Sheet H7-8 4-09-1 Sectors

1. Find the area of the sectors OAB in the diagrams below (to 3sf):

(a) (b) (c)

(d) (e) (f) 2. Find the length of the curved edge AB in the above diagrams (to 3sf). 3. The area of the minor sector AOB shown below is 60 2m . (a) Find an expression for the area of the minor sector AOB in terms of x.

(b) By putting this expression equal to 60 2m , find the angle x (to 1dp). (c) Find the exact length of the minor arc AB (by using “Ans” on calculator).

PTO

O

A

8m

x

B

O

A

17mm 95°

B

O

A

35cm

130°

B

O A

9m °30

B

O

A

2m

B

O

A

15cm 9°

B

O

A

°180

B

10mm



Sheet H7-8 4-09-1 Sectors (cont.)

4. In the diagram below AB = 85cm, X is the midpoint of AB, and O is the centre of the circle (a) Show that °=∠ 62.3AOX (to 1dp) (by using trigonometry). (b) Find the angle AOB (to 1dp). (c) Find the length OX (to 3sf). (d) Find the area of triangle AOB (to 3sf). (e) Find the area of the sector AOB (to 3sf). (f) Find the area of the shaded segment (to 3sf).

O

A

48cm

B

85cm

X



O

A

B X 50

10cm

Sheet H7-9 4-09-2 Sectors 1. In the diagram shown below,

(a) Find (to 3sf) the area of the minor sector OAB. (b) Find the length of the major arc AB.

The minor arc is cut out and thrown away. OA is joined to OB and so a cone shape is formed. (c) By using your answer to part (b) find the exact value of the radius of the circular base of this cone. (d) What is the height of this cone? (Hint : Use Pythagoras’ Theorem). 2. The area of the minor sector AOB shown below is 50cm2 .

Find the angle x (to 1dp).

3. A circle has radius 5cm with centre O. A and B are points on the circumference such that the angle AOB is °120 . Find the area of the sector AOB (to 3sf). 4. A circle has radius 6cm with centre O. A and B are points on the circumference such that the angle AOB is °30 . Show that the area of the sector AOB is exactly 3π . 5. A circle has radius 8cm with centre O. A and B are points (with AB being a horizontal line) on the circumference such that the angle AOB is °40 . X is the midpoint of AB.

(a) Use trigonometry to find the length of AX and hence show that the length of AB is 5.47cm (to 3sf).

(b) Use trigonometry to show that the height of the triangle AOB is 7.52cm (to 3sf). (c) Find the area of the triangle AOB (to 3sf). (d) Find the area of the sector AOB (to 3sf). (e) Show that the area enclosed between the line AB and the circle is 1.77 2cm (to 3sf).

6. A circle has radius 10cm with centre O. A and B are points on the circumference such that the angle AOB is °50 . The point X lies on line AB such that OX is perpendicular to AB.

(a) Find OX and AX (to 3sf). (b) Find the area of the triangle AOX (to 3sf). (c) Find the area of the sector AOB (to 3sf). (d) Show that the area enclosed between the lines AX, XB and the circle is 19.0 2cm (to

3sf).

PTO

O A

B

10cm x

O

A

B5cm °72



Sheet H7-9 4-09-2 Sectors (cont.) 7. Two circles of radius 10cm have centres P and Q where PQ is horizontal and has length 16cm. The two circles intersect at A and B. The line AB intersects the line PQ at a point X. Y is the point on the circle with centre P which lies on the line PQ.

(a) Write down the lengths of PX and AP. (b) Use trigonometry to find the angle APX (to 1dp). (c) Write down the length of AX, using Pythagoras. (d) Find the area of the triangle APX. (e) Find the area of the sector APY (to 3sf). (f) Hence show that the area enclosed between the two circles 32.8 2cm (to 3sf).

8. Two circles of radius 13cm have centres P and Q where PQ is horizontal and has length 24cm. Use the method of question 7 to show that the area which is common to both circles is 13.4 2cm (to 3sf). 9. (a) Find, in terms of π , l and x, the value of the dark shaded area.

(b) If the minor arc is cut away and OA is joined to OB then a cone shape is formed. Find, in terms of l and x, the radius, R, of the circular base of the cone. (c) Hence find (in terms π , l and R) the area of the curved surface area of the cone.

O

A

B

x

l

P Q

B

A

X Y



Sheet H7-10 4-09-3 Area and Volume

Volume of cone is 13 πr 2h Volume of cylinder is πr 2h

Volume of sphere is 34

3 rπ Curved surface area of cone is lrπ

Curved surface area of cylinder is 2 rhπ Surface area of sphere is 4πr2

Give answers in terms of π where appropriate.

31000 litres 1m= , 31000cm 1 litre= 1. A cylindrical metal tin (with top and base), has a radius of 5cm and a height of 4cm.

(a) How much metal is needed to cover this tin? (b) What volume will it hold? (c) What is the ratio of surface area to volume?

2. A large cylindrical water tank has a radius of 2m and a height of 3m. How much water (in

litres) will it hold? 3. Which of the two closed cylinders has the largest surface area? 4. Calculate the volumes of the above shapes. 5. Which has the larger volume – a cuboid measuring 12cm by 4cm by 10cm or a cylinder

with radius 4cm and height 10cm? 6. How many cylinders of radius 5cm and height 2cm would be filled from the water contained

in a cylinder of radius 10cm and height 8cm? 7. A cylinder of radius 5cm and height 12cm is filled up with water. The water is then poured

into a cone with height 4cm so that it fills the cone exactly. What is the base radius of the cone?

8. A cone has height 24cm and base radius 3cm is filled with ice cream. How many spheres of

radius 3cm could be filled by the ice cream? PTO

20cm

50cm

40cm

11cm A B



Sheet H7-10 4-09-3 Area and Volume (cont.) 9. A circle has radius 12cm. A sector AOB is cut out where O is the centre and A and B are

points on the circle such that the angle AOB is °60 .

(a) Find the area of the sector. (b) Find the perimeter of the arc AB. (c) The sides OA and OB are then joined together to form a cone as shown below.

Find the base radius of the cone using part (b). (d) Hence show that the formula lrπ for the curved surface area of a cone does give us the

right area. (e) Find the height (leaving your answer as a square root) of the cone.

10. A circle has radius 25cm. A sector AOB is cut out where O is the centre and A and B are points on the circle such that the angle AOB is θ .

(a) Show that the perimeter of the arc AB is 536πθ .

(b) The sides OA and OB are then joined together to form a cone. If the height of the cone is 24cm then find the base radius.

(c) Hence find the perimeter of the base of the cone and use this to find θ . 11. A circle has radius 6cm. A sector AOB is cut out where O is the centre and A and B are

points on the circle such that the angle AOB is θ .

(a) Show that the area of the sector AOB is 10πθ 2cm and the perimeter is

30πθ cm.

(b) The sides OA and OB are then joined together to form a cone. If the height of the cone is 3cm then find the base radius.

(c) Hence find the curved area of the cone and use this to find θ .

l

r

A O

B

°60

12cm

A O

B

θ

25cm



Sheet H7-11 4-09-4 Area and Volume 31000 litres 1m= , 31000cm 1 litre=

1. A cylindrical tin with radius 5cm and height 12cm is made of metal. It has both a top and a

bottom. (a) How much metal (to 3sf) is needed to cover this tin? (b) What volume (to 3sf) will it hold?

2. A large cylindrical water tank has a radius of 1m and a height of 2.5m. How much water (to

the nearest 10 litres) will it hold? 3. Calculate the total surface areas of the following shapes (to 3sf): 4. Calculate the volumes of the above shapes (to 3sf). 5. A cone has base radius 17cm and height 67cm. Find the volume of the cone (to 3sf). 6. Calculate the volume (to 3sf) of a cone whose radius is 3.2m and whose height is 6.1m. 7. What is the surface area (to 3sf) of a cricket ball if its diameter is 10cm? 8. What is the volume (to 3sf) of a netball whose radius is 15cm? 9. A cylindrical glass of radius 4cm and height 9cm is filled up with water. The water is then

poured into a upturned cone of base radius 5cm and height 15cm until it is filled up. How much water (to 2sf) will be left in the glass?

10. A company makes spherical and cubical ice holders. When frozen both shapes are full.

What is the radius (to 3sf) of the spherical container if it holds as much as the cubical container of side length 13mm?

11. An ice cream scooper is designed to create spheres of radius 2.4cm. It is used to serve ice

cream from a container measuring 20cm by 15cm by 13cm. If the scooper always picked up perfect spheres of radius 2.4cm then how many scoops (to the nearest whole number) would this container hold?

12. A 3 litre pot of paint is used to paint the surface of a large sphere and the instructions say

that one litre will cover 2m5 . What is the maximum radius of the sphere (to 3sf)? 13. A cone of height 45cm has to hold 3 litres of water. What is the least possible value (to 3sf)

of the base radius? Volume of cone is 1

3 πr 2h . Curved surface area of cone is lrπ where l is the slant height

Surface area of sphere is 4πr2 . Volume of sphere is 34

3 rπ .

3m

1.5m

(a) (b)

83cm

37cm



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Sheet H7-12 4-09-5 Area and Volume 1. An apple (assume that it is spherical) of radius 4cm is cut in half. What area of cling film

will be needed to cover completely one half of the apple (to 3sf)?

2. A cylinder of radius 6cm and height 8cm has a hemisphere (i.e. half a sphere) stuck on top of it (also of radius 6cm). (a) What is the ratio of the volume of cylinder to the volume of the hemisphere? (b) What is the volume of this shape (to 3sf)?

3. A cylinder of height 7cm and radius 6cm is filled up with water. If the water is then poured

into a cone with radius 5cm and exactly fills up the cone what is the height of the cone? 4. A sphere has volume 4m3 - what is its radius (to 3sf)? 5. A cylindrical glass has a radius of 3cm and holds 3360cm of water when full. Find the

height of the glass (to 3sf). 6. A cone has curved surface area 6cm 2 and radius 1cm. What is slant height (to 3sf) of the

cone? 7. The total surface area of a closed cylinder is 175cm2 (including its two ends). Its radius is

4cm. What is the height of this cylinder (to 3sf)?

Volume of cone is 13 πr 2h .

Curved surface area of cone is lrπ where l is the slant height


3 rπ .

PTO

l

1cm

4cm



Sheet H7-12 4-09-5 Area and Volume (cont.)

8. Using the fact that 3.14 (to 3sf)π = to find, without using a calculator, the volume (to 3sf) of a cylinder with radius 5cm and height 4cm.

9. A security light switches itself on when it detects movement in a given area, which is a

sector of a circle. The angle of this sector is °120 and the radius is 12m. What is the area (to 3sf) of this sector?

10. A cake has diameter 35cm and height 8cm. A slice of cake is cut so that the angle of the

slice is 40° . Calculate the volume of this piece of cake (to 3sf). 11. A cylindrical container has a base radius of 10cm. Water is poured into the cylinder to a

height of 5cm. A heavy solid sphere of radius 3cm is placed into the water so that it rests on the bottom of the cylinder. By how much does the height of the water increase?




3 rπ .



Sheet H7-13 4-09-6 Area and Volume 1. A railway tunnel is constructed in the shape of a

hollow cylinder. It is 1km long and has a radius of 3m. A gravel bed is laid in order to support the track.

A cross section of the tunnel is shown opposite, with the shaded area representing the gravel.

AB represents the horizontal surface of the gravel. X is the midpoint of AB.

(a) Show that AOB is °60 . (b) Show that (to3sf) m602.OX= . (c) Find the area (to 3sf) of the triangle OAB. (d) By considering the area of the sector OAB and your answer to part (b), find (to 3sf) the shaded area. (e) Hence find (to 3sf) the volume of gravel required for the tunnel. (f) Find (to 3sf) the length of the major arc ACB. (g) If the wall of the tunnel above the gravel level is to be painted, find (to 3sf) the surface area to be painted.

2. A 1.5 litre bottle of champagne is

placed in a cylindrical ice bucket whose diameter is 16cm and whose height is 25cm. The bucket is then completely filled with cold water so that the bottle stays completely immersed. Ignoring the volume of the glass of the bottle, what is the volume of water (to 3sf) used to fill the bucket?

(NB: 1 litre = 1000 3cm ) 3. 1.5 litres of champagne is mixed with an equal volume of orange juice in a hemispherical

bowl which is exactly filled by the drink. Find the radius (to 3sf) of the bowl. 4. Some wine is poured into the conical glass opposite so that it has a

depth of 8cm and diameter 5cm (as shown).

(a) Find the volume of the wine in the form 3

mπ .

(b) Find the volume of a spherical ice cube of radius 1 cm form

3nπ .

(c) The ice cube is then put into the glass. Assuming that the ice floats such that 25% of its volume is above the surface of the liquid, calculate the rise (to the nearest mm) in level of the drink in the glass after the ice cube is put in.




3 rπ .

B60°

3m

C

X

25cm

16cm

8cm

5cm



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Sheet H7-14 4-09-7 Area and Volume 1. A golf ball is covered in a coating of width 3mm. If it has a radius of 2cm before it is

covered then find the volume of the coating (to 3sf). 2. A cylindrical tank. It is fully covered with a insulating foam which is 15cm thick. If the tank has height 1m and base radius 0.8m before it is covered then what is the volume of foam used (to 3sf)? 3. In the above diagram O is the centre of the circle with radius 5cm. A and B are points on the circle and 8AB = cm.

(a) Find the area of the circle (to 3sf). (b) By splitting the triangle OAB into two triangles with a vertical line and by then using

trigonometry, find the angle AOB (to 1dp). (c) Calculate the area of the sector AOB (to 3sf). (d) Find the height of the triangle AOB (to 3sf). (e) Find the area of the triangle AOB (to 3sf). (f) Calculate the area (to 3sf) of the segment enclosed between the line AB and the circle.

4. In the above diagram O is the centre of the circle with radius 13m. A and B are points on the circle, X is the midpoint of AB and 12OX = m.

(a) Find the angle AOX and hence the angle AOB (to 1dp). (b) Calculate the area of the sector AOB (to 3sf). (c) Find the length of AX. (d) Find the area of the triangle AOB. (e) Calculate the area (to 2sf) of the segment enclosed between the line AB and the circle.

Volume of cone is 1

3 πr 2h . Curved surface area of cone is lrπ where l is the slant height


3 rπ . PTO

A B

O 5cm

8cm

A B

O13m

12m

X



Sheet H7-14 4-09-7 Area and Volume (cont.) 5. A cylindrical pipe lies on it side. The pipe is 2m long and has a radius of 25cm. It has some water lying in the pipe such that the maximum depth of the water is 18cm.

(a) Draw a diagram of this pipe and the water. (b) Find the cross sectional area (to 3sf) of the water in the pipe. (c) Find the volume (to 3sf) of water in the pipe.

6. A cylindrical pipe lies on it side. The pipe is 3m long and has a radius of 10cm. It has some water lies in the pipe such that the maximum depth of the water is 16cm.

(a) Draw a diagram of this pipe and the water. (b) Find the cross sectional area (to 3sf) of the water in the pipe. (c) Find the volume (to 3sf) of water in the pipe.




3 rπ .



Sheet H7-15 4-10-1 Similar Shapes 1. A hardware shop sells two similar buckets. One holds 5 litres and is 30cm tall, the other is

35cm tall. How much (to 3sf) does the larger bucket hold? 2. A fabric shop sells similar rugs. A customer buys a rug which is 2m from one corner to the

opposite corner and which has an area of 3.6 2m . What will the area be of a similar rug which is 3m from one corner to the opposite corner?

3. In a photograph of a tractor, the tractor is 18cm long and its area is 120cm2 . On the

negative, which is similar to the photograph, the tractor is 45mm long. What is the area of the tractor in the negative?

4. A certain vase is 25cm high and holds 31500cm of water. How much water (to 3sf) will a

similar vase hold if it is 18cm high? 5. A company produces non-spherical rubber balls of various sizes. The ball with a height of

7cm has a volume of 50 3cm . Find (to the nearest 3cm ) the volume of a similar ball whose height is 6cm.

6. A shop sells bars of soap in various sizes, all of which are similar to each other. If the bar

which is 5cm long has a volume of 80 3cm then what is the volume (to 3sf) of the bar which is 8cm long?

7. Two football pitches are similar to each other. One is 80m long and the groundsman needs

£15 worth of weed killer to cover the pitch. How much weed killer (to the nearest pence) is required for the pitch which is 60m long?

8. A container of height 8cm is made from 200cm2 of metal and holds 300cm6 of liquid. If a

similar container has a height of 15cm then: (a) how much metal will needed to make it? (to 3sf) (b) how much liquid will it hold? (to 3sf)

9. A piece of string of length 15cm encloses an area of 8cm2 . If a string of length 25cm

encloses an area of a similar shape, what is the area of this shape (to 3sf)?

8cm 15cm



Sheet H7-16 4-10-2 Similar Shapes

1. Two milk bottles are similar. The larger one is 20cm tall and has a volume of 1.5 litres. How much (to 3sf) will the smaller bottle hold if it is 15cm tall?

2. Two picture frames are similar. The smaller one is 18cm long and has an area of 32 2cm . If

the larger one is 24cm long then find its area (to 3sf). 3. The cost of similar bottles of milk is proportional to the volume. A bottle which has a

radius of 36mm costs 75p. (a) What is the price (to the nearest p) of the bottle with a radius of 45mm? (b) What is the radius (to 3sf) of the bottle which costs £1?

4. In the (non right-angled) triangle shown below BE is parallel to CD.

(a) Draw the triangles ABE and ACD and hence show that 15mmx = . (b) If the area of triangle ABE is 1000 2mm then find the area of the triangle ACD.

5. Two books are similar. The smaller one has a surface area of 2cm64 and a width of 7cm.

What is the surface area (to 3sf) of the larger book if it had width 9cm? 6. Two round cakes are similar. One has radius 10cm and weighs 1.5kg. What is the radius

(to 3sf) of the other cake if it weighs 2kg? 7. In the diagram below AB= 5cm, BC= 4cm and the area of the triangle ABE is 23 2cm .

Given that BE is parallel to CD find the area of the triangle ACD. 8. Two water containers are similar. One holds 5 3m and the other holds 12 3m . If the height

of the smaller one is 1.2m what is the height (to 3sf) of the larger one? 9. Two pictures are similar. The area of one is 54 2cm and the other is 216 2cm . If the length

of the larger one is 18cm then find the length of the smaller one.

A

B

C

E D

A

B C

E D

50mm 40mm 60mm

x




1. A factory makes two footballs and they charge a fixed price per square metre of leather that is used to cover the football. If they charge £12 for the football of radius 15cm how much do they charge for a football of radius 12cm?

2. The cost of a fizzy drink needed to fill a cylindrical can of height 10cm is 35p. How much

(to the nearest pence) does is cost to fill a similar can of height 13cm with the same fizzy drink?

3. An art shop charges £10 for the area of glass that is needed to cover a painting of length

50cm. How much would it charge to cover a similar painting of length 70cm with the same sort of glass?

4. A shop sells two sizes of plastic cylindrical containers which are similar to each other. The

larger container has a diameter of 28cm and a volume of 20 litres. What is the diameter (to 3sf) of the container which holds 15 litres?

5. An egg box of height 54mm has a volume of 3cm300 . What is the height (to 3sf) of a

similar egg box if its volume is 350cm5 ? 6. The two triangles shown below are similar.

The area of the larger triangle is 2m6 and its base is 5m. How long (to the nearest cm) is the base of the smaller triangle if its area is 2m1 ?

7. Two text books are similar. One has a weight of 600g, the other has a weight of 1kg. If the

smaller book has length 20cm, find the length (to 3sf) of the larger book. 8. Two buildings in photographs are similar. The height of one is 10cm and the height of the

other is 13cm. If the area of the building in the smaller photograph is 45 2cm then find the area of the building in the larger photograph.

9. Two rugby balls are similar. The length of one is 10cm and has a volume of 150 3cm . Find

the volume (to 3sf) of the ball that has a length of 18cm. 10. Two posters are similar. The poster with diagonal 60cm has an area of 450 2cm . Find the

diagonal of the poster with area 800 2cm . PTO

28cm

15 litres 20

litres



Sheet H7-17 4-10-3 Similar Shapes (cont.) 11. When 3cm250 of water is poured into the cone shown below it reaches a height of hcm.

When a further 38cm120 of water is poured in, the level of water rises by 8cm.

(a) Draw two diagrams, marking on the volumes and the heights of the water.

(b) Write down an equation involving h to show that 8 95

hh+

= .

(c) Solve this equation to find the value of h. 12. Two milk churns are similar. One has a radius of 20cm, a surface area of 3600 2cm and a

volume of 8 litres. The smaller one has a surface area of 900 2cm . Find the volume of the smaller churn.




1. Two inkpots are similar. One has a volume of 30 3cm and a surface area of 60 2cm . The other has a surface area of 240 2cm . (a) If the length of the shorter pot is 5cm then find the length of the larger pot. (b) Use this to find the volume of the larger pot.

2. Two rugby balls are similar. One has a volume of 5000 3cm and a surface area of 1500 2cm . The other has a volume of 625 3cm . (a) Find the ratio of the volumes of the two balls. (b) Calculate the ratio the lengths of the two balls. (c) Use this to find the ratio of the surface areas of the two balls. (d) Hence find the surface area of the smaller ball.

3. Two containers are similar. One has a volume of 6000 3cm and a surface area of 2500 2cm . The other has a volume of 1000 3cm and surface area A.

(a) Fill in the blanks in the following : 2

3.........

A⎛ ⎞

= × ⎜ ⎟⎜ ⎟⎝ ⎠

.

(b) Use this to show that A = 757 2cm (to 3sf).

4. Two boxes are similar. One has a volume of 900 3cm and a surface area of 500 2cm . The other has a surface area of 800 2cm and volume V.

(a) Fill in the blanks in the following:3

......

...V

⎛ ⎞= × ⎜ ⎟⎜ ⎟

⎝ ⎠.

(b) Use this to find V (to 3sf).

5. Three layers of wedding cake are similar. The middle layer has a surface area of 3600cm2

and a mass of 5kg. (a) What is the mass (to 3sf) of the bottom layer if its surface area is 8000cm2 ? (b) What is the surface area (to 3sf) of the top layer if its mass is 3kg?

6. Two cuboids are similar. One has volume 6 3m and the other has volume 11 3m . If the

surface area of the smaller one is 22 2m what is the surface area (to 3sf) of the larger one?

PTO



Sheet H7-18 4-10-4 Similar Shapes (cont.) 7. Two containers are similar. One has height 36cm and volume 11 litres, the other has height

29cm. What is its volume (to 3sf)? 8. Two shapes are similar, with areas of 13m2 and 25m2 respectively. If the smaller shape

has a length of 6m, what is the corresponding length in the other shape (to 2sf)?



Sheet H7-19 4-10-5 Similar Shapes 1. A cuboid with volume 340cm2 has surface area 2cm236 . What is the volume (to 3sf) of a

similar cuboid whose surface area is 232cm4 ? 2. A plant pot has a volume of 3.5 litres and a surface area of 250cm8 . What is the volume

(to 3sf) of a plant pot whose surface area is 200cm12 ? 3. Two masses of 25kg and 32kg respectively are of similar shape. If the surface area of the

smaller mass is 2.2m4 then what is the surface area (to 3sf) of the larger mass? 4. X and Y are two geometrically similar solid shapes.

The total surface area of shape X is 450 cm2 and the total surface area of shape Y is 800 cm2. Given that the volume of shape X is 1350 cm3, calculate the volume of shape Y.

5. The diagram below represents two wine glasses. Glass A is in the shape of a cone, height 8cm and radius rcm. Glass B is in the shape of a hemisphere with radius 4cm.

Glass A Glass B

(a) Find the volume of wine (to 3sf) in Glass B when it is full. (b) If Glass A holds 325cm1 of wine when it is full then find r (to 3sf). (c) If only 3cm64 of wine is poured into Glass A then explain why the shape it makes is

similar to the shape of the wine in glass A initially. (d) Using similar shapes find the depth of the wine in the glass if only 3cm64 of wine is

poured into Glass A. (e) Using similar shapes find, as a fraction, the factor by which the surface area of wine

exposed to the air is reduced when Glass A goes from being full to having only 3cm64 of wine in it.

Volume of cone is 13 πr 2h . Volume of sphere is

343 rπ .

rcm

8cm

4cm



Sheet H7-20 5-01a-1 Vectors

1. If A, B and C are the points (2, 5), (4, -2) and (-1, 1) respectively then find the following: (a) AB (b) BC (c) BA (d) AC (e) an equation connecting AB , BC and AC .

(f) the co-ordinates of the point E where 65

AE ⎛ ⎞= ⎜ ⎟⎝ ⎠

.

(g) the co-ordinates of the point D such that ABDE is a parallelogram.

2. If ⎟⎟⎠

⎞⎜⎜⎝

⎛=

32

a , ⎟⎟⎠

⎞⎜⎜⎝

⎛=

15

b and ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

43

c then find the following:

(a) 3a (b) 5b

(c) ( )ba +21

(d) cb − (e) ab 35 + (f) c3ba 42 +−

3. ABCD is a parallelogram where A, B, C and D have position vectors ⎟⎟⎠

⎞⎜⎜⎝

⎛43

, ⎟⎟⎠

⎞⎜⎜⎝

⎛69

, ⎟⎟⎠

⎞⎜⎜⎝

⎛97

and

⎟⎟⎠

⎞⎜⎜⎝

⎛71

respectively and E is the midpoint of AB. Find the following:

(a) OE (b) AB (c) DE (d) EC (e) the co-ordinates of X, the midpoint of AC (f) the co-ordinates of Y, the midpoint of BD

4. ABCD is a parallelogram such that p=AB and q=BC . Find the following in terms of p

and q: (a) CD (b) AD (c) AC (d) AM where M is the midpoint of AB (e) AN where N is the midpoint of AC (f) AP where P is the point along AC which is twice as far from A as from C (P is between A and C)




Do this on a double page in your book. Put the set of axes on the left hand page and answer all questions on the right hand page.

1. (a) Draw a set of axes from –8 to 8 on both axes (1cm per unit) and mark on it the points

A (-6, -6) and C (-1, -2).

(b) Find the co-ordinates of B where ⎟⎟⎠

⎞⎜⎜⎝

⎛=

14

AB .

(c) Find the column vector BC . (d) ABCD is a parallelogram (lettered anticlockwise). Find the co-ordinates of D and

draw the parallelogram on your axes. 2. (a) On the set of axes used in question 1 draw the triangle PQR (lettered anticlockwise)

where P is (2, 0), Q is (6, 2) and R is (4, 6). (b) If PQ=a and QR=b then find a and b as column vectors.

(c) Show that ba +=PR . (d) Mark on your diagram the points X, Y and Z which are the midpoints of PQ, QR and

RP respectively. (e) Find the following in terms of a and b:

(i) XQ

(ii) QY

(iii) PY (iv) XY

(f) To which side of the triangle PQR is XY parallel. (g) How can you prove this using any of the above results?

3. (a) On the set of axes used in questions 1 & 2 draw the triangle LMN (lettered

anticlockwise) where L is (-7, 1), M is (-1, 3) and N is (-4, 7). (b) If LM=p and MN=q then find p and q as column vectors.

(c) Express NL in terms of p and q. (d) Mark on your diagram the point S which is the midpoint of LM. (e) The point T lies on LN and cuts it in the ratio 2:1. Show that T is the point (-5, 5). (f) Find the following in terms of p and q:

(i) SM (ii) NT (iii) ST

(g) Draw the image of LMN when it is translated by the vector ⎟⎟⎠

⎞⎜⎜⎝

⎛−87

.




1. ABCD is a parallelogram (labelled anticlockwise) with p=AB and q=AD . M is the midpoint of AD, N is the midpoint of AB and R is one quarter of the way along AC from A. (a) Find the following in terms of p and q (in the form …p +…q):

RNMR

ARAC

ANAM

)vi()v(

)iv()iii(

)ii()i(

(b) Hence show that M, N and R all lie on a straight line. (c) Find the ratio MR:MN.

2. ABC is a triangle (labelled anticlockwise) with p=AB and q=AC . X, Y and Z are the midpoint of AB, BC and CA respectively. (a) Find the following in terms of p and q (in the form …p +…q):

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) (viii)

BC BY

XB XY

YC YZ

AZ XZ

(b) Hence show that XZ is parallel to BC. (c) What is the ratio of XZ:BC?

3. ABCD is a parallelogram (labelled anticlockwise) with r=AB and s=AD . X is two thirds

of the way along BD from B and Y is one third of the way along AD from A. (a) Find the following in terms of r and s (in the form …r +…s):

YXAY

AXBX

ACBD

)vi()v(

)iv()iii(

)ii()i(

(b) Hence show that YX is parallel to AC. (c) What is the ratio of YX:AC?

4. ABC is a triangle (labelled anticlockwise) with p=AB and q=AC . D,E and F are the

midpoints of AB, BC and CA respectively. G is two thirds of the way along AE from A. (a) Find the following in terms of p and q (in the form …p +…q):

AGAE

BEBC

)iv()iii(

)ii()i(

(b) Show that ( )pq 231 −=BG and that ( )pq 22

1 −=BF . (c) Hence show that B, G and F all lie on a straight line. (d) Find expressions for CG and CD to show that C, G and D all lie on a straight line. (e) The point G is the point of intersection of which three lines?




1. ABCD is a parallelogram. P, Q, R and S are the midpoints of AB, BC, CD and DA respectively. u=AB and v=BC . (a) Express the following terms of u and v (i.e. in the form …u +…v):

BDSQ

SRAR

AQAC

)vi()v(

)iv()iii(

)ii()i(

(b) Use the above to show that SR is parallel to AC. (c) Find the ratio SR:AC. (d) Express BD in the form…u +…v. (e) Express AX in the form…u +…v where X is the midpoint of BD. (f) Express AY in the form…u +…v where Y is one third of the way along BD from B.

2. ABC is a triangle such that AB = p and BC = q .

(a) The point X lies on the line AB produced (i.e. X lies further from A than B does) so that AB:BX = 2:1. Find AX in terms of p and q.

(b) If Y lies on BC, between B and C so that BY:YC = 1:3, find AY in terms of p and q. (c) Find AC in terms of p and q. (d) Given that Z is the midpoint of AC, find AZ in terms of p and q (you may find it

easier to calculate AC ) (e) Find XY and YZ in terms of p and q. (f) Show that XYYZ = and hence explain why X, Y and Z all lie on the same straight

line.

3. ABCD is a parallelogram (labelled anticlockwise) with p=AB and q=AD . M lies on AB

such that AM:MB=3:1 and N is on AD such that AN:ND=3:5 and N. The point X is on AC such that AX:XC=1:3. (a) Find the following in terms of p and q (i.e. as …p+…q).

(i) (ii) (iii)

(iv) (v) (vi)

AM AN AC

AX MX XN

(b) Hence show that M, N and X all lie on a straight line. (c) Find the ratio MX:NX.



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