surface area 6 and volume - miss cannings maths links · 10/01/2012 · 174 new signpost...

6Surface Areaand Volume

173

How much paint

will I need?

Chapter Contents6:01 Review of surface area MS4·2, MS5·2·26:02 Surface area of a pyramid MS5·3·16:03 Surface area of a cone MS5·3·1Investigation: The surface area of a cone6:04 Surface area of a sphere MS5·3·1Investigation: The surface area of a sphereFun Spot: How did the raisins win thewar against the nuts?6:05 Volume of a pyramid MS5·2·2

Investigation: The volume of a pyramid6:06 Volume of a cone MS5·2·26:07 Volume of a sphere MS5·2·2Investigation: Estimating your surfacearea and volume6:08 Practical applications of surface

area and volume MS5·2·2, MS5·3·1Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically

Learning OutcomesMS4·2 Calculates surface area of rectangular and triangular prisms and volume of right prisms

and cylinders.MS5·2·2 Applies formulae to find the surface area of right cylinders and volume of right pyramids, cones

and spheres and calculates the surface area of and volume of composite solids.MS5·3·1 Applies formulae to find the surface area of pyramids, right cones and spheres.

Working Mathematically Stages 5·3·1–51 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reflecting

174 New Signpost Mathematics

Enhanced 10 5.1–5.3

6:01

Review of Outcomes

MS4·2,

MS5·2·2

Surface AreaIn Year 9, the surface areas of prisms, cylinders and composite solids were calculated by adding the areas of the faces (or surfaces).

The following formula may be needed.

Worked examples1 Find the surface area 2 Find the surface area 3 Find the surface area

of this prism. of this cylinder. of this solid.

Solutions1 Surface area = 2LB + 2LH + 2BH

= 2 × 2·9 × 3·7 + 2 × 2·9 × 2·1 + 2 × 3·7 × 2·1

= 49·18 m2

2 For a cylinder:

Surface area = curved surface area + area of circles

Surface area = 2πrh + 2πr2

= (2π × 4·35 × 6·8) + (2 × π × 4·352)

= 304·75 m2 (correct to 2 dec. pl.)

3 Surface area = 2 × (14 × 14) − π × 42 + 4 × (14 × 6) + 2π × 4 × 4 + π × 42

= 728 + 32π= 829 cm2 (to nearest cm2)

Area formulae1 square: A = s2 8 Surface area of a rectangular prism:

2 rectangle: A = LB A = 2LB + 2LH + 2BH

3 triangle: A = bh

4 trapezium: A = h(a + b)

5 parallelogram: A = bh 9 Surface area of a cylinder:

6 rhombus: A = xy A = 2πrh + 2πr2

7 circle: A = πr2

L

H

B

h2�r

r

12---

12---

12---

2·1 m

2·9 m3·7 m

6·8 m

8·7 m

14 cm14 cm

The hole is 4 cm deep

8 cm

6 cm

6·8 m

8·7 m4·35 m

175Chapter 6 Surface Area and Volume

Find the surface area of the following prisms.a b

c d

Find the surface area of the following solids.a b c

In each of the following questions use Pythagoras’ theorem tocalculate the unknown length x, correct to two decimal places, and then calculate the surface area.a b

Find the surface area of the following solids. All measurements are in metres.a b c

Find the surface area of the following solids. All measurements are in centimetres.a b c

Exercise 6:01 Surface area review MS4·2, MS5·2·21 Find the surface area of the

following rectangular prisms.a

2 Find the surface area of the following triangular prisms.a

3 Find the surface area of the following cylinders.a

12

3

8

38

6

56

Foundation Worksheet 6:01

1

15 cm

7 cm

8 cm

10 cm

14 cm12 cm

8 cm

16 cm10 cm

18 cm

14 cm

33·6 cm

8 cm15 cm

5 cm

6 cm8 cm

3

4

2

6 cm

8 cm

9 cm7 cm

8 cm6 cm

■ Caution!c2 = a2 + b2

Right-angled triangles

a

b

c

3

6 cm

9 cm12 cm

x cm

x cm

11 cm

14 cm

7 cm7 cm

4

7

12

14

109

33

3

89

410 6

8 9

12

5

10

12

4

45 3

3

5

8

126

8

88

20

3

176 New Signpost Mathematics Enhanced 10 5.1–5.3

6:02 Surface Area of a Pyramid Outcome MS5·3·1

1 Are the triangular faces of a square pyramid congruent?

2 Are the triangular faces of a rectangular pyramid congruent?

3 Find x

4 The net of a squarepyramid is shown. Find the area of the net.

Worked examplesCalculate the surface area of the following square and rectangular pyramids.

1 2 3

Solutions1 Surface area = (area of square) + 4 × (area of a triangular face)

= (10 × 10) + 4 ×

= 100 + 260

= 360 cm2

ABCDE is a rectangular pyramid.

Find: Find the area of:

5 OX 9 ΔEBC

6 EX 10 ΔABE

7 OY

8 EY

E

10

12

16

O

D

A B

X

C

Y

Prep Quiz 6:022·

4 cm

x cm

1·8 cm

8 6

To calculate the surface area of a pyramid with a polygonal base, we add the area of the base and the area of the triangular faces.

10 cm

10 cm

13 cmE D

C

A

B

O

AO = 8 cmBC = 6 cm

T LM = 8 cmMN = 6 cmTO = 7 cm

N

ML

O

P

■ In right square pyramids, all the triangular faces are congruent. This simplifies the calculation of the surface area.

10 13×2

------------------⎝ ⎠⎛ ⎞


2 As the perpendicular height of the triangular face is not given, this must be calculated.

In the diagram, AM is the perpendicular height ofthe face.

In ΔAOM

AM2 = AO2 + OM2 (Pythagoras’ theorem)

= 82 + 32 (Note: OM = CD)

= 64 + 9

= 73

AM =

Surface area = (area of square) + 4 × (area of a triangular face)

=

=

= 138·5 cm2 (correct to 1 dec. pl.)

3 The perpendicular heights TA and TB mustbe calculated, as these are not given.

In ΔTOA,TA2 = AO2 + OT2

= 32 + 72

= 58∴ TA = cm

In ΔTOB,TB2 = BO2 + OT2

= 42 + 72

= 65∴ TB = cm

Now,

Surface area = (area of rect. LMNP) + 2 × (area of ΔTMN) + 2 × (area of ΔTLM)

=

=

=

= 157·3 cm2 (correct to 1 dec. pl.)

AO = 8 cmBC = 6 cm

B

E D

OCM

A12---

73

6 6×( ) 4 6 73×2

-------------------⎝ ⎠⎛ ⎞×+

36 12 73+

T

N

B

MAL

OP

4 cm3 cm

7 cmOB = 1

2LM

OA = 12MN

58

65

■ In right rectangular pyramids, the opposite triangular faces are congruent.

LM MN×( ) 2 MN TB×2

-----------------------⎝ ⎠⎛ ⎞ 2 LM TA×

2----------------------⎝ ⎠

⎛ ⎞×+×+

8 6 2 6 65××2

----------------------------- 2 8 58××2

-----------------------------+ +×

48 6 65 8 58+ +


The following diagrams represent the nets of pyramids.Calculate the area of each net.a b

Calculate the surface area of the following square and rectangular pyramids.a b c

Use Pythagoras’ theorem to calculate the perpendicularheight of each face and then calculate the surface areaof each pyramid. Give the answers in surd form.a b

Calculate the surface area of the following pyramids. Give all answers correct to one decimal place where necessary.a b c

Exercise 6:02 Surface area of a pyramid MS5·3·11 Calculate the area of each net.

a

2 Find the surface area of each square or rectangular pyramid.a

6

5

66

10


1

6 cm

6 cm

6 cm

6 cm

6 cm 6 cm

12 cm

12 cm

17·86 cm

18 cm32 cm

2

6 cm

6 cm

10 cm7·5 cm

7·5 cm

9·5 cm

4 cm

2 cm

17 c

m

20 cm

3

E

C

B

X

A

DO

AB = 10 cmEO = 12 cm

T

LM = 10 cmTO = 10 cm

L M

X

N

O

P

4

A

F

GH

OE

EFGH is a rectangle.AO = 8 cm, HG = 12 cm,GF = 9 cm.

M

Z

YX

O

W

WXYZ is a square.MO = 3 cm, XY = 11 cm.

E D

C

A

OB

BCDE is a rectangle.AO = 4 cm, ED = 10 cm,DC = 6 cm.


Find the surface area of:a a square pyramid, base edge 6 cm, height 5 cmb a rectangular pyramid, base 7 cm by 5 cm, height 10 cm

Find the surface area of the following solids. Give all answers correct to three significant figures.a b c

Find the surface area of a pyramid that hasa regular hexagonal base of edge 6 cm and a height of 8 cm.

A square pyramid has to have a surface area of 2000 cm2.If the base edge is 20 cm, calculate:a the perpendicular height, x cm, of one of the

triangular facesb the perpendicular height, h cm, of the pyramid

6:03 Surface Area of a Cone Outcome MS5·3·1

1 Area = ?

2 Circumference = ?

3 Simplify:

4 Simplify:

5

6

M

G

C

BA

ED O

H

F

ABCD is a square.AB = 5 m, MO = 3 m,CG = 2 m

A

E

B

CO

D

F

AO = OB = 9 cmCD = DE = FE = FC = 6 cm

4 cm

2 cm

3·5

cm

7

x cmh cm

20 cm20 cm

8

What fraction of a circle is each of the following sectors?

5 6 7

8 9 10 Evaluate πrs ifr = 3·5 and s = 6·5.Answer correct to1 dec. pl.

O O120° O

3 cm circumference = 9 cm

O

6 cm

O

2πr

circumference = 2πs

Prep Quiz 6:03

s

2πr2πs---------

rs-- πs2×


A cone may be thought of as a pyramid with a circular base.Consider a cone of slant height s and base radius r.• Imagine what would happen if we cut along a straight line

joining the vertex to a point on the base.• By cutting along this line, which is called the slant height,

we produce the net of the curved surface. The net of thecurved surface is a sector of a circle, radius s.

The surface area of a cone comprises two parts: a circle and a curved surface. The curved surface is formed from a sector of a circle.• This investigation involves the making of two cones

and the calculation of their surface area.Step 1Draw a semicircle of radius 10 cm.Step 2Make a cone by joining opposite sides of the semicircle, as shown below.Step 3Put the cone face down and trace the circular base.Measure the diameter of this base.Step 4Calculate the area of the original semicircle plus the area of the circular base.This would be the total surface area of the cone if it were closed.

• Repeat the steps above, making the original sector a quarter circle of radius 10 cm.What is the surface area of a closed cone of these dimensions?

Investigation 6:03 The surface area of a cone

The centre of AB on the

semicircle is the point of the cone.

1 2 3 4

BA10 cm 10 cm

sticky tape

A B

dBA

10 cm 10 cm

A= 12π(10)2

r A= πr 2

r

sh

A

B

2πr

s

A

B

2πr

A

BB

s

curved surface

r

base

If you bring the two ‘B’s

together, the sector

will bend to form

a cone.


To calculate the area of a sector, we must find what fraction it is of the complete circle. Normally this is done by looking at the sector angle and comparing it to 360°, but it can also be done by comparing the length of the sector’s arc to the circumference of the circle. Hence, if the sector’s arc length is half the circumference of the circle, then the sector’s area is half the area of the circle. (See Prep Quiz 6:03.)

∴ Area of sector =

=

= πrsNow, since the area of the sector = area of the curved surface,

curved surface area = πrs

Find the curved surface area of the followingcones, giving answers in terms of π.a b

Worked examples1 Find the surface area of a cone with a radius of 5 cm and a slant height of 8 cm.

2 Find the surface area of a cone with a radius of 5 cm and a height of 12 cm.

Solutions1 Surface area = πrs + πr2

= π × 5 × 8 + π × 52

= 40π + 25π= 65π cm2

= 204·2 cm2

(correct to 1 dec. pl.)

length of sector’s arccircumference of circle--------------------------------------------------------- area of circle×

2πr2πs--------- πs2×

Surface area of a cone:surface area = πrs + πr2

where r = radius of the cone ands = slant height of the cone

Note: s h2 r2+=

s

r

h

2 First the slant height mustbe calculated.Now, s2 = 52 + 122

(Pythag. theorem)= 169

∴ s = 13surface area = πrs + πr2

= 65π + 25π= 90π cm2

= 282·7 cm2

(correct to 1 dec. pl.)

12

5

s

■ The height of a cone is the perpendicular height.

Exercise 6:03Surface area of a cone MS5·3·11 For each cone shown, find the:

i radius ii height iii slant heighta b

2 For each of the cones in question 1 find:i the curved surface area ii the surface area

3 Use Pythagoras’ theorem to find the slant height if:a radius = 3 cm; height = 4 cmb diameter = 16 cm; height = 15 cm

8 10

6O

7·512·5

20O


1

8 cm

10 cm

10 cm

10 cm


c d e

Find the surface area of each of the cones in question 1 giving all answers in terms of π.

Calculate the surface area of the followingcones, giving all answers in terms of π.a radius 8 cm and height 6 cmb radius 1·6 m and height 1·2 mc diameter 16 cm and height 15 cmd diameter 1 m and height 1·2 m

In each of the following, find the surface area of the cone correct to four significant figures.a radius 16 cm and height 20 cm b radius 5 cm and slant height 12 cmc radius 12·5 cm and height 4·5 cm d diameter 1·2 m and height 60 cme diameter 3·0 m and slant height 3·5 m

Find the surface area of the following solids. Give all answers correct to one decimal place.a b c

A cone is to be formed by joining the radii of thesector shown. In the cone that is formed, find:a the slant heightb the radiusc the perpendicular height

a A cone with a radius of 5 cm has a surface area of 200π cm2. What is the perpendicular height of the cone?

b A cone cannot have a surface area greater than 1000π cm2. What is the largest radius, correct to one decimal place, that will achieve this if the slant height is 20 cm?

12 cm

16 cm

40 cm

20 cm

1·2 m

r = 40 cm

2

Don’t forget,

use the slant

height, not the

vertical height,

of the cone.

3

4

5

A

D O

CB

AB = 12 cmBO = 12 cmBC = 5 cmOD = 10 cm

9 cm

7 cm7 cm

12 cm

10 cm

Note: This is half a cone

15 cm

15 cm 120°

6

7


6:04 Surface Area of Outcome MS5·3·1

a Sphere

Carry out the experiment outlined below todemonstrate that the reasoning is correct.

Step 1Cut a solid rubber ball or an orange intotwo halves.The faces of the two hemispheres are identical circles.

Step 2Push a long nail through the centre of a hemisphere, as shown.

Step 3Using thick cord, cover the circular faceof one of the hemispheres as shown, carefully working from the inside out.Mark the length of cord needed. Call this length A.Length A covers the area of the circle, ie πr2.

Step 4Put a second mark on the cord at a point that is double the length A.The length of the cord to the second mark is 2A.2A covers the area of two identical circles, ie 2πr2.

Step 5Turn the other hemisphere over and use the cord of length 2A to cover the outside of the hemisphere.It should fit nicely.It seems that 2A covers half of the sphere.It would take 4A to cover the whole sphere.ie the surface area of a sphere = 4A

Surface area = 4πr2

It is impossible to draw

the net of a sphere in

two dimensions...

Investigation 6:04 The surface area of a sphere

1

2 3 4 5


The proof of the formula of the surface area of a sphere is beyond the scope of this course.

Find the surface area of a sphere with:a radius = 5 cm b radius = 7·6 cm c radius = 3·2 md diameter = 18 cm e diameter = 1·6 m f diameter = 8000 kmGive all answers as exact answers (multiples of π) and also as approximations correct to three significant figures.

Calculate the exact surface area of a hemisphere with:a a radius of 12 cm b a diameter of 12 cm

Calculate the surface area of each solid correct to two decimal places.a b c

Worked examples1 Find the surface area of a 2 Find the surface area of the

sphere of diameter 12 cm. hemisphere shown here.

Solutions1 Diameter = 12 cm 2 S = area of curved surface + area of circle

∴ Radius = 6 cm = 2πr2 + πr2

Now, S = 4πr2 = 3πr2

= 4 × π × 62 = 3π × 72

= 144π cm2 = 147π cm2

= 452 cm2 = 461·8 cm2

(correct to 3 sig. figs) (correct to 1 dec. pl.)

r

The surface area of a sphere is given by the formula:SA = 4πr2

where SA is the surface area and r is the radius.

Don’t forget the flat surface

when finding the surface area

of a hemisphere.

7 cm

Exercise 6:04

1

2

3

8 cm19·4 m

22 m


A cylinder is 3 m long and has a radius of 80 cm. A hemispherical cap is placed on each end of the cylinder. Calculate the surface area of the solid correct to four significant figures.

A sphere is to have a surface area of 200 cm2. Find its radius correct to one decimal place.

Calculate the surface area of each of the following solids, correct to three significant figures.(All measurements are in centimetres unless stated otherwise.)a b c

4

5

6

5

10

A

B O

O is the centre of the hemisphere

8

8

16

O D

C

A

B

BC = 0·6 mAB = 0·8 mOD = 3·0 mAO = 4·0 m

• The dimetrodon was a dinosaur that had a large sail to absorb and dissipate heat efficiently. Dimetrodons of different sizes had a slightly different shape. Larger specimens had proportionally larger sails. These sails varied according to the volume rather than the surface area or the length of the creature.

Being cold blooded, they needed to remove or absorb heat according to their body mass (or volume).


Work out the answer to each question and put the letter for that part in the box that is above the correct answer.

Simplify where possible:

B 8x + x C 8x ÷ x

D 8x × x E 8x − x

H xy + yx E 2x − 2

E (x + 1)2 − 1 E 3(2x − 4) − 6x

T 3x × 3x T −2x − 5x

Solve:

G H E H

I x2 = 9 I 0·3x = 3 J L

For the formula A = πrs:

L what is the subject?

M how many variables are used?

N what is the value of π correct to four significant figures?

Find the value of b2 − 4ac if:

N a = 2, b = 10, c = 5 N a = 3, b = −2, c = 4

O a = 1, b = −1, c = 4 S a = 6, b = −3, c = −1

Simplify:

S (a−1)−2 S a−1 × a1 E (2a2)3 T

T E T U

Y 6a + 3a ÷ 3

Fun Spot 6:04 How did the raisins win the war against the nuts?

x3--- 3= x

3--- 1

3---= 3x 1

3---= 3

4---x 3

8---=

x 9= x x2---=

x16

a4--- a

5---+ a2

5----- 5

a---× 3a a

4---÷ a 3a

16------–

x8

x = 7x 7a a2

9x

x =

81

a 8

9x2

−12

8x2

12

x =

1

x =

3

−15

x =

10 60 −7x

2x −

2

−44 1

8a6

33 2xy

x2 +

2x

x =

0 A

x =

±3

3·14

2

x =

9

1 2---

13a

16--------

- 1 9---

9a

20------


6:05 Volume of a Pyramid Outcome MS5·2·2

• Photocopy the nets below onto light cardboard.• Use these nets (with tabs) to construct an open rectangular prism of length 4·2 cm,

breadth 4·2 cm and height 2·8 cm and an open pyramid of length 4·2 cm, breadth 4·2 cm and height 2·8 cm.(Note: to produce this pyramid, each triangle must have a height of 3·5 cm.)

© Pearson Australia. Reproduction of this page from New Signpost Mathematics Enhanced 10 Stage 5.1–5.3 may be made for classroom use.

Investigation 6:05 The volume of a pyramid

tab

tab

tab

tabta

b

The paper, the net and the box

• Fill the pyramid with sugar or sand and tip the contents into the rectangular prism. How many times must this bedone before the prism is full?

• Complete: The volume of the pyramid = of the volume of the prism.

1------

Greatest volume


A pyramid is named according to the shape of its base.• BCDE is the base of the pyramid.• A is the apex of the pyramid.• AO is the height of the pyramid. It is sometimes called

the altitude.• Investigation 6:05 demonstrates that the volume

of a pyramid is one-third of the volume of a prism withthe same base area and height.

Worked examples1 Find the volume of a rectangular pyramid that has a base 6·2 cm long and 4·5 cm wide and

a height of 9·3 cm.

2 A pyramid has a hexagonal base with an area of 12·6 cm2. If the height of the pyramid is 7·1 cm, calculate its volume.

3 Calculate the volume of the pyramids pictured.a b

Solutions1 V = Ah 2 V = Ah 3 a V = Ah

A = 6·2 × 4·5 cm2 A = 12·6 cm2 A = 6·5 × 6·5 cm2

h = 9·3 cm h = 7·1 cm h = 12 cm

∴ V = × 6·2 × 4·5 × 9·3 cm3 ∴ V = × 12·6 × 7·1 ∴ V = × 6·5 × 6·5 × 12

= 86·49 cm3 = 29·82 cm3 = 169 cm3

3 b V = Ah

A = cm2

h = 10·5 cm

∴ V = ×

= 782·04 cm3

D

CB

E

O

A

A

hThe volume of all pyramids is given by the formula:

V = Ah

where V = volume, A = area of base and h is the height of the pyramid.

13---

B

C D

EO

A AO = 12 cmBC = 6·5 cmEB = 6·5 cm

BM

C

A

D

O

BC = 26·6 cmDM = 16·8 cmAO = 10·5 cm

13--- 1

3--- 1

3---

13--- 1

3--- 1

3---

D

CB M26·6 cm

16·8 cm

The base is triangular. It’s drawn below.

Area of base =

= cm2

BC DM×2

-----------------------

26·6 16·8×2

-----------------------------

13---

13---

26·6 16·8×2

----------------------------- 10·5×


Calculate the volume of the square and rectangular pyramidsdrawn below.a b

a Calculate the volume correct to two decimal places of asquare pyramid with a base edge of 3·25 m and a heightof 6·3 m.

b A rectangular pyramid is 16·2 cm high. Its base is 5·8 cm long and 7·5 cm wide. Find its volume.

a A pyramid has a triangular base with an area of 4·32 m2. Find the volume of the pyramid if it has a height of 2·5 m.

b Find the volume of a hexagonal pyramid of height 15 cmif the hexagonal base has an area of 6·2 cm2.

a A square pyramid has a volume of 18 000 cm3. If theside length of the square is 30 cm, what is its height?

b A pyramid has a 12 cm square as its base. How high mustthe pyramid be if it is to have a volume of 500 cm3?

c A square pyramid with a height of 120 cm has a volumeof 64 000 cm3. What is the area of its base and the length of the side of the square?

d A rectangle that is twice as long as it is wide forms the base of a rectangular pyramid that is 60 cm high. If the volume of the pyramid is 4000 cm3, what are the dimensions of the rectangle?

Calculate the volume of the triangular pyramids below. The base of each pyramid has been drawn separately alongside it.a b

Calculate the volume of the pyramids drawn below.a b

Exercise 6:05 Volume of a pyramid MS5·2·21 In each of the following, calculate the

volume of the prism and then divide by 3 to find the volume of the pyramid.

a b

2 For each of the following pyramids, find the value of A and h and then use the

formula to find the volume.

a b

6

66

56

10

V Ah3

-------=

63

8

3

5

3


1

A

O

12 cm12 cm

AO = 10 cm

O

AO = 6·3 cmA

6·5 cm

14·2 cm

2

3 In question 4 you substitute

into the formulae and then

solve the equation.

V = 13 Ah 18000 = 13 � 30 � 30 � h h = ?

4

5

AO = 12·0 cmC

B D O

A

C

D B

12·3 cm 12·2 cm

AO = 16·2 cmC

B D O

A

15 cmB D

C

8 cm

Watch these two—they’re tricky.

Change centimetres to metres

before doing any calculations.

6

85 cm

85 cm

O

A

AO = 1·2 m1·6 mB C

D

A

E

O 1·2 m

AO = 95 cm


Calculate the volume of the following solids.a b c

d e

a Calculate the volume of a square pyramid if it has abase area of 64 cm2 and the distance from the apex to a corner of the base is 15 cm.

b Calculate the volume of a pyramid that has a heightof 8 cm. The base of the pyramid is a regular hexagon with a side length of 6 cm. The base is shown in the diagram.

A tetrahedron is a triangular pyramid in which each face is an equilateral triangle. Calculate the volume of a tetrahedron that has all its edges 6 cm in length. Hint: You will need to know some geometry and trigonometry. A diagram of the pyramid and its base are drawn below.

7

7·5 m5·2 m

1·6 m4·

1 m

A

L

D C

M

O

B

ABCD is a square.AB = 12 cmLO = MO = 10 cm

L

G

C

BA

D

E FM

H

O

ABCD and EFGH are squares.AB = 20 cm, EF = 10 cmMO = 15 cm, LM = 15 cm

12·6 cm 21·5 cm

The height of the pyramidis 8·5 cm. (Answer correctto 3 sig. figs.)

3·0 m3·0 m

6·0 m

7·5 m7·5 m A

O

B

The solid in

part c is a

truncated

pyramid. It’s

volume is

calculated by

subtraction.

AO = 4·5 mBO = 3·0 m

A B

C F

E D

OO is thecentre ofthe base

8

9

C

A

B

O

D

B C

A

O

You’ll have to be wide

awake for this one,


6:06 Volume of a Cone Outcome MS5·2·2

Just as the cylinder could be thought of as a ‘circular prism’, so the conecan be thought of as a ‘circular pyramid’.

The volume of a cone is one-third of the volume of a cylinder with the same base area and height.

Find the volume of the following cones correct to two decimal places.a b c

Worked examples1 Find the volume of a cone 2 Use Pythagoras’ theorem to

with a radius of 6·2 cm and calculate the height, h, correcta height of 5·8 cm. Give to three decimal places andyour answer correct to then use this value to calculatethree significant figures. the volume correct to three

significant figures.

Solutions1 V = πr2h

r = 6·2 cm

h = 5·8 cm

∴ V = × π × (6·2)2 × 5·8

= 233 cm3 (correct to 3 sig. figs)

h

r

The volume of a cone is given by the formula:

V = πr2h

where r is the radius of the circular base and h is the height.

13---

h cm

3·6 cm

7·5 cm

2 By Pythagoras’ theorem

h2 + 3·62 = 7·52

h2 = 7·52 − 3·62

h =

= 6·580 (correct to 3 dec. pl.)

Now V = πr2h

r = 3·6

h = 6·580

∴ V = × π × 3·62 × 6·580

= 89·3 cm3 (correct to 3 sig. figs)

7·52 3·62–

13---

13---

13---

13---

Exercise 6:06

1

5·9 m

3·8 m

2·7 m

2·5 m

17·5 cm

21·3 cm


a A cone has a base radius of 5·2 cm and a height of 7·8 cm. Calculate its volume correct to two significant figures.

b The diameter of the base of a cone is 12·6 cm. If the cone has a height of 15·3 cm, find its volume correct to three significant figures.

c A cone has a base diameter of 2·4 m and a height of 45 cm. Calculate its volume to the nearest tenth of a cubic metre.

Use Pythagoras’ theorem to calculate h, correct to three decimal places, and then use this value to calculate the volume correct to three significant figures.a b c

A right-angled triangle with sides of 5, 12 and 13 cmis rotated to form a cone. What is the volume of the cone if it is rotated about:a the 12 cm side? b the 5 cm side?

a A cone has a radius of 10 cm. It has a volume of 1000 cm3. Calculate the height of the cone correct to one decimal place.

b A cone with a height of 20 cm has a volume of 1500 cm3. Calculate the radius of the cone correct to one decimal place.

c It is noticed that the height of a cone is twice its radius and that the cone’s volume is exactly 1000 cm3. Calculate the dimensions of the cone correct to one decimal place.

d A conical flask has a radius of 10 cm and a height of 10 cm. The contents of the cone are emptied into a cylinder with a radius of 5 cm. How high must the cylinder be to hold the contents of the cone?

Calculate the volume of the following solids. Give all answers correct to three significant figures.a b c

The sector shown is formed into a cone by joining its two radii. Calculate the volume of the cone correct to the nearest whole number.

2

3

5·6 mh m

2·5 mh cm

2·7 cm

15·5 cm

4·8 mh m

3·6 m

4

5

6

17·5 cm

8·5 cm

5·8 cm

7·6 m

A D

B

C O

8·2 m

8·2 m3·8 m

B

C O

A

D

AB = 4·8 cmBO = 6·4 cmBD = 3·6 cmOC = 4·8 cm

240°

12 cm

7


6:07 Volume of a Sphere Outcome MS5·2·2

As with the pyramid and cone, the formula for the volume of a sphere is hard to prove.

Here the formula is just stated without proof.

• Describe how you could estimate your surface area using a suitable roll of material.

• The volume of a solid can be established by placing it in a container of water. The amount of water that it displaces is equivalent to the volume of the solid.Describe how you could use this principle, which was discovered by Archimedes, to calculate your volume.

Worked examples1 Find the volume of a sphere that has a radius of 5·20 cm.

Give your answer correct to three significant figures.2 If the diameter of a sphere is 3·6 m, calculate the volume

of the sphere correct to one decimal place.3 If the earth is considered to be a sphere of radius 6378 km,

find its volume correct to four significant figures.

Solutions1 V = πr3 2 Diameter = 3·6 m

= × π × (5·20)3 ∴ Radius = m

= 589 cm3 (correct to 3 sig. figs) = 1·8 m

3 V = πr3 ∴ V = πr3

= × π × 63783 = × π × (1·8)3

= 1·087 × 1012 km3 = 24·4 m3

(correct to 4 sig. figs) (correct to 1 dec. pl.)

Investigation 6:07 Estimating your surface area and volume

The volume of a sphere is given by the formula:

V = πr3

where r is the radius of the sphere.

43---

Do you remember how

to use the xy button

on your calculator?

43---

43---

3·62

--------

43--- 4

3---

43--- 4

3---


Find the exact volumes of the following spheres:a radius = 3 cm b diameter = 4 mc radius = 3·6 m d diameter = 4·8 m

Find the volumes of the following solids. Give the answers correct to three significant figures.a sphere, radius 1·2 m b sphere, diameter = 25·6 cmc hemisphere, radius 3·15 cm d hemisphere, diameter = 2·40 m

Calculate the volume of the following solids, correct to three significant figures.a b c

Calculate the volume of a spherical shell that has an inner radius of 5 cm and an outer radius of 6 cm. Give the answer in terms of π.

a A sphere has a volume of 2000 cm3. Calculate its radius correct to one decimal place.b What is the smallest radius a sphere can have if it has to have a volume greater than

5000 cm3? Round your answer up to the nearest whole number.c A sphere has to have a volume of 1 m3. What is the radius of the sphere? Round your

answer up to the nearest centimetre.

Exercise 6:07

1

2

3

8·4 cm

12·6 cm

20·0

cm

6·9 cmO

4·0 cm

4·0 cm

2·0 cm

4

5

• Why is gas stored in spherical tanks? Estimate the volume of gas held in one of these tanks.


6:08 Practical Applications Outcomes MS5·2·2, MS5·3·1

of Surface Area and Volume

A swimming pool is rectangular in shape and has uniform depth.It is 12 m long, 3·6 m wide and 1·6 m deep. Calculate:a the cost of tiling it at $75/m2

b the amount of water in litres that needs to be added to raise thelevel of water from 1·2 m to 1·4 m

The tipper of a trick is a rectangular prism in shape. It is 7 m long, 3·1 m wide and 1·6 m high.a Calculate the volume of the tipper.b If the truck carries sand and 1 m3 of sand

weighs 1·6 tonnes, find the weight of sand carried when the truck is three-quarters full.

c Calculate the area of sheet metal in the tipper.

Worked exampleA buoy consists of a cylinder with two hemispherical ends, as shown in the diagram. Calculate the volume and surface area of this buoy, correct to one decimal place.

SolutionSince the two hemispheres have the same radius, they will form a sphere if joined.

= πr3 + πr2h

= π(1·1)3 + π × 1·12 × 2·4

= 14·7 m3 (correct to 1 dec. pl.)

= 4πr2 + 2πrh= 4π × 1·12 + 2 × π × 1·1 × 2·4= 31·8 m2 (correct to 1 dec. pl.)

1·1 m

2·4 m

1·1 m

■ The height of the hemisphere is equal to its radius.

This means the

hemispheres and the

cylinder have the

same radius.

Volumeof buoy

volume ofsphere⎝ ⎠

⎛ ⎞ volume ofcylinder⎝ ⎠

⎛ ⎞+=

43---

43---

Surfacearea

surface areaof sphere

curved surfacearea of cylinder

+=

Exercise 6:08

■ 1 m3 = kL1

2

7 m

3·1

m

1·6 m


A large cylindrical reservoir is used to store water. The reservoir is 32 m in diameter and has a height of 9 m.a Calculate the volume of the reservoir to the nearest cubic metre.b Calculate its capacity to the nearest kilolitre below its maximum capacity.c In one day, the water level drops 1·5 m. How many kilolitres of water does this represent?d Calculate the outside surface area of the reservoir correct to the nearest square metre.

Assume it has no top.

Assuming that the earth is a sphere of radius 6400 km, find (correct to 2 sig. figs):a the volume of the earth in cubic metresb the mass of the earth if the average density

is 5·4 tonnes/m3

c the area of the earth’s surface covered by water if 70% of the earth is covered by water.

A bridge is to be supported by concrete supports. Calculate the volume of concrete needed for each support.

The supports are trapezoidal prisms, as pictured inthe diagram.Give the answer to the nearest cubic metre.

A house is to be built on a concrete slabwhich is 20 cm thick. The cross-sectionalshape of the slab is shown in the diagram.Calculate the volume of concrete neededfor the slab.

A steel tank is as shown in the diagram.

Given that the dimensions are externaldimensions and that the steel plate is 2 cmthick, calculate the mass of the tank if thedensity of the steel is 7·8 g/cm3. Give theanswer correct to one decimal place.

3

4

5

6·4 m

6·2 m

1·8 m

3·6 m

6·8 m

8·5 m

4·7 m

10·8 m

2·1 m

2·1 m

6

710·5 m

2·4 m3·5 m


Calculate the volume of a concrete beam that hasthe cross-section shown in the diagram. The beam is 10 m long.

Calculate the mass of this beam if 1 m3 of concreteweighs 2·5 tonnes.

The large tank in the photo consists of two cones and a cylinder. If the diameter of the cylinder is 5·2 m andthe heights of the bottom cone, cylinderand top cone are 2·8 m, 8·5 m and1·8 m respectively, calculate the volume of the tank correct to one decimal place.

A storage bin has been made from a square prism and asquare pyramid. The top 1·8 m of the pyramid has beenremoved. Calculate the volume of the bin.

A glasshouse is in the shape of a squarepyramid. Calculate the area of the fourtriangular faces to the nearest squaremetre if the side of the square is 20 mand the height of the pyramid is 17 m.

8

1800 mm

700 mm1300 mm

300 mm

300 mm

100 mm

100 mm

9

10 3·6 m

1·2 m

A

B

CAB = 4·5 mBC = 3·6 m

11


a The solid shown is known as a frustrum. It is formed by removing the top part of the cone.

i By comparing the values of tan θ in two different triangles, find the value of x.

ii Find the volume of the frustrum.

b A storage bin for mixing cement is formed fromtwo truncated cones (frustrums). Calculate thevolume of this bin.

composite solid• A solid that is formed

by joining simple solids.

prism• A solid that has two identical ends

joined by rectangular faces.

pyramid• A solid that has a base from

which triangular faces rise to meet at a point.

slant height (of a cone)• The distance from a point on the

circumference of the circular base to the apex of the cone.

surface area• The sum of the areas of the faces or surfaces

of a three-dimensional figure (or solid).

volume• The amount of space (cubic units) inside a

three-dimensional shape.

1 m

2·7 m

2·5 m

12

1 m

2·7 m

x m

2·5 m

�

�

2·0 m

2·5 m

2·7

m2·

4 m

1 m

1 2 3 Literacy in Maths Maths terms 6

slantheight

Maths terms 6


• Each part of this test has similar items that test a certain question type.• Errors made will indicate areas of weakness. • Each weakness should be treated by going back to the section listed.

These questions can be used to assess all of outcome MS5·3·1 and parts of outcome MS5·2·2.

Diagnostic Test 6 Surface Area and Volume

1 Calculate the surface area of the following pyramids.

2 Calculate the surface area of the following cones. Give answers correct to one decimal place.a b c

3 Calculate the surface area of:a a sphere of radius 5 cm, correct to 2 dec. pl.b a sphere of diameter 16·6 cm, correct to 2 dec. pl.c a hemisphere of radius 3 cm, correct to 2 dec. pl.

4 Calculate the volume of the following solids.

5 Calculate the volume of the following solids correct to one decimal place.a b c

6 Calculate the volume of the following solids correct to one decimal place:a a sphere of radius 5 cm b a sphere of diameter 8·6 cmc a hemisphere of diameter 15 cm

Section6:02

6:03

6:04

6:05

6:06

6:07

12 cm

12 cm10 c

m

a

8 cm

6 cm

12·65 cm

12·3

6 cm

b

10 cm

10 cm

12 c

m

c

5 cm

6·5 cm

1·8 cm

2·4

cm

2·6 m

2·6

m

5·4 m5·4 m

4·6 m

a

8·4 m5·4 m

5·3 m

b

16·4 cm 8·7 cm

7·6 cm

c

2·6 cm

6·5 cm 6·1 cm

4·3 cm

2·6 cm

12·3 cm

surface area 6 and volume - miss cannings maths links · 10/01/2012 · 174 new signpost...

Documents