and volume 6
TRANSCRIPT
NSME_10_5-3_SB_06_CD.fmChapter Contents
6:02
6:03
Surface area of a cone MS5·3·1
Investigation: The surface area of a cone 6:04
Surface area of a sphere MS5·3·1
Investigation: The surface area of a sphere Fun Spot: How did the raisins win the war against the nuts? 6:05
Volume of a pyramid MS5·2·2
Investigation: The volume of a pyramid 6:06
Volume of a cone MS5·2·2
6:07
Investigation: Estimating your surface area and volume 6:08
Practical applications of surface area and volume MS5·2·2, MS5·3·1
Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically
Learning Outcomes
MS4·2
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.
W
orking
M
athematically
S
tages
Questioning,
2
Surface Area
In 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 examples
3
Find the surface area of this prism. of this cylinder. of this solid.
Solutions
1
3
)
Area formulae 1 square: A = s2 8 Surface area of a rectangular prism:
2 rectangle: A = LB A = 2LB + 2LH + 2BH
3 triangle: A = bh
5 parallelogram: A = bh 9 Surface area of a cylinder:
6 rhombus: A = xy A = 2πrh + 2πr2
7 circle: A = πr2
6·8 m
8·7 m
8 cm
6 cm
Sam ple
pa ge
a b
c d
a b c
In each of the following questions use Pythagoras’ theorem to calculate 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, MS5·2·2 1 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
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 square pyramid is shown. Find the area of the net.
Worked examples Calculate the surface area of the following square and rectangular pyramids.
1 2 3
Solutions 1 Surface area = (area of square) + 4 × (area of a triangular face)
= (10 × 10) + 4 ×
5 OX 9 ΔEBC
6 EX 10 ΔABE
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
T LM = 8 cm MN = 6 cm TO = 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
177Chapter 6 Surface Area and Volume
2 As the perpendicular height of the triangular face is not given, this must be calculated.
In the diagram, AM is the perpendicular height of the face.
In ΔAOM
= 64 + 9
= 73
AM =
=
=
= 138·5 cm2 (correct to 1 dec. pl.)
3 The perpendicular heights TA and TB must be calculated, as these are not given.
In ΔTOA, TA2 = AO2 + OT2
= 32 + 72
= 42 + 72
Now,
=
=
=
AO = 8 cm BC = 6 cm
B
------------------- ×+
In right rectangular pyramids, the opposite triangular faces are congruent.
LM MN×( ) 2 MN TB× 2
----------------------- 2 LM TA×
----------------------------- 2 8 58×× 2
Sam ple
pa ge
178 New Signpost Mathematics Enhanced 10 5.1–5.3
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 perpendicular height of each face and then calculate the surface area of 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 MS5·3·1 1 Calculate the area of each net.
a
2 Find the surface area of each square or rectangular pyramid. a
6
5
7·5 cm
9·5 cm
T
L M
O E
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
O B
BCDE is a rectangle. AO = 4 cm, ED = 10 cm, DC = 6 cm.
Sam ple
pa ge
179Chapter 6 Surface Area and Volume
Find the surface area of: a a square pyramid, base edge 6 cm, height 5 cm b 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 has a 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 faces b 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:
H
F
ABCD is a square. AB = 5 m, MO = 3 m, CG = 2 m
A
E
B
AO = OB = 9 cm CD = DE = FE = FC = 6 cm
4 cm
2 cm
3· 5
8
What fraction of a circle is each of the following sectors?
5 6 7
8 9 10 Evaluate πrs if r = 3·5 and s = 6·5. Answer correct to 1 dec. pl.
O O 120° O
O
180 New Signpost Mathematics Enhanced 10 5.1–5.3
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 the curved 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 1 Draw a semicircle of radius 10 cm. Step 2 Make a cone by joining opposite sides of the semicircle, as shown below. Step 3 Put the cone face down and trace the circular base. Measure the diameter of this base. Step 4 Calculate 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
sticky tape
A B
d BA
If you bring the two ‘B’s together, the sector
will bend to form a cone.
Sam ple
pa ge
181Chapter 6 Surface Area and Volume
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 =
=
= πrs Now, since the area of the sector = area of the curved surface,
curved surface area = πrs
Find the curved surface area of the following cones, giving answers in terms of π. a b
Worked examples 1 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.
Solutions 1 Surface area = πrs + πr2
= π × 5 × 8 + π × 52
= 40π + 25π = 65π cm2
(correct to 1 dec. pl.)
length of sector’s arc circumference of circle --------------------------------------------------------- area of circle×
2πr 2πs --------- πs2×
Surface area of a cone: surface area = πrs + πr2
where r = radius of the cone and s = slant height of the cone
Note: s h2 r2+=
s
r
h
2 First the slant height must be calculated. Now, s2 = 52 + 122
(Pythag. theorem) = 169
= 65π + 25π = 90π cm2
12
5
s
The height of a cone is the perpendicular height.
Exercise 6:03 Surface area of a cone MS5·3·1 1 For each cone shown, find the:
i radius ii height iii slant height a 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 cm b diameter = 16 cm; height = 15 cm
8 10
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 following cones, giving all answers in terms of π. a radius 8 cm and height 6 cm b radius 1·6 m and height 1·2 m c diameter 16 cm and height 15 cm d 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 cm c radius 12·5 cm and height 4·5 cm d diameter 1·2 m and height 60 cm e 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 the sector shown. In the cone that is formed, find: a the slant height b the radius c 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
height, not the vertical height, of the cone.
3
4
5
A
CB
AB = 12 cm BO = 12 cm BC = 5 cm OD = 10 cm
9 cm
15 cm
6:04 Surface Area of Outcome MS5·3·1
a Sphere
Carry out the experiment outlined below to demonstrate that the reasoning is correct.
Step 1 Cut a solid rubber ball or an orange into two halves. The faces of the two hemispheres are identical circles.
Step 2 Push a long nail through the centre of a hemisphere, as shown.
Step 3 Using thick cord, cover the circular face of 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 4 Put 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 5 Turn 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...
1
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 the level 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 example A 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.
Solution Since the two hemispheres have the same radius, they will form a sphere if joined.
= πr3 + πr2h
= 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.
+=
196 New Signpost Mathematics Enhanced 10 5.1–5.3
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 metres b 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 in the diagram. Give the answer to the nearest cubic metre.
A house is to be built on a concrete slab which is 20 cm thick. The cross-sectional shape of the slab is shown in the diagram. Calculate the volume of concrete needed for the slab.
A steel tank is as shown in the diagram.
Given that the dimensions are external dimensions and that the steel plate is 2 cm thick, calculate the mass of the tank if the density of the steel is 7·8 g/cm3. Give the answer correct to one decimal place.
3
4
5
Sam ple
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197Chapter 6 Surface Area and Volume
Calculate the volume of a concrete beam that has the cross-section shown in the diagram. The beam is 10 m long.
Calculate the mass of this beam if 1 m3 of concrete weighs 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 and the heights of the bottom cone, cylinder and top cone are 2·8 m, 8·5 m and 1·8 m respectively, calculate the volume of the tank correct to one decimal place.
A storage bin has been made from a square prism and a square pyramid. The top 1·8 m of the pyramid has been removed. Calculate the volume of the bin.
A glasshouse is in the shape of a square pyramid. Calculate the area of the four triangular faces to the nearest square metre if the side of the square is 20 m and the height of the pyramid is 17 m.
8
11
198 New Signpost Mathematics Enhanced 10 5.1–5.3
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 from two truncated cones (frustrums). Calculate the volume of this bin.
composite solid • A solid that is formed
by joining simple solids.
joined by rectangular faces.
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
slant height
199Chapter 6 Surface Area and Volume
• 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 cm c a hemisphere of diameter 15 cm
Section 6:02
4·6 m
5·3 m
7·6 cm
4·3 cm
2·6 cm
12·3 cm
200 New Signpost Mathematics Enhanced 10 5.1–5.3
1 Calculate the volume and surface area of the pyramid shown.
2 A cone has a diameter of 16 cm and a height of 15 cm. Calculate the surface area of the cone in terms of π.
3 Calculate the surface area of a hemisphere that has a diameter of 16 cm. Give your answer correct to two significant figures.
4 A spherical shaped tank is to hold 100 m3. What radius to the nearest millimetre will give a volume closest to 100 m3?
5 Calculate the volume of the solid pictured. Give the answer correct to three significant figures.
6 Calculate the surface area of the solid shown. Give the answer correct to three significant figures.
7 A tank for holding liquid chemicals consists of a cylinder with two hemispherical ends as shown in the diagram. Calculate its volume (correct to 3 sig. figs.)
8 An octahedron is a double pyramid with all its edges equal in length. Calculate the surface area and volume of an octahedron with all its edges 6 cm in length.
Revision Chapter 6 Revision Assignment
9 cm
9 cm
6 cm
6 cm
• Engineers solve many surface area and volume problems in the design and construction of buildings.
Sam ple
pa ge
201Chapter 6 Surface Area and Volume
1 The value of a library is depreciated at a rate of 15% pa. If the library is presently valued at $800 000, what will its value be after 4 years?
2 ΔABC is equilateral with a side of length a units. AT, BS and RC are perpendicular heights of the triangle that meet at X. Find the lengths: a BX b XT
3 ABCD is a right triangular pyramid. Its base is an equilateral triangle of side units and the edges AB, AC and AD are all 6 units long. Find the height, AX and the volume of the pyramid.
4 Three positive whole numbers are multiplied in pairs. The answers obtained are 756, 1890 and 4410. What are the numbers?
5 The dot indicates the position of one chess queen on a chess board. How many more queens can you place on the board so that none of the queens threatens another?
6 One large sheet of paper was ruled up and folded. It was then cut along the fold shown in the diagram on the top to form an 8-page booklet. On the diagrams to the right, put the page number on each quarter (as has been done for 2 and 4).
Revision Chapter 6 Working Mathematically
A
Sam ple
pa ge
6:02
6:03
Surface area of a cone MS5·3·1
Investigation: The surface area of a cone 6:04
Surface area of a sphere MS5·3·1
Investigation: The surface area of a sphere Fun Spot: How did the raisins win the war against the nuts? 6:05
Volume of a pyramid MS5·2·2
Investigation: The volume of a pyramid 6:06
Volume of a cone MS5·2·2
6:07
Investigation: Estimating your surface area and volume 6:08
Practical applications of surface area and volume MS5·2·2, MS5·3·1
Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically
Learning Outcomes
MS4·2
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.
W
orking
M
athematically
S
tages
Questioning,
2
Surface Area
In 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 examples
3
Find the surface area of this prism. of this cylinder. of this solid.
Solutions
1
3
)
Area formulae 1 square: A = s2 8 Surface area of a rectangular prism:
2 rectangle: A = LB A = 2LB + 2LH + 2BH
3 triangle: A = bh
5 parallelogram: A = bh 9 Surface area of a cylinder:
6 rhombus: A = xy A = 2πrh + 2πr2
7 circle: A = πr2
6·8 m
8·7 m
8 cm
6 cm
Sam ple
pa ge
a b
c d
a b c
In each of the following questions use Pythagoras’ theorem to calculate 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, MS5·2·2 1 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
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 square pyramid is shown. Find the area of the net.
Worked examples Calculate the surface area of the following square and rectangular pyramids.
1 2 3
Solutions 1 Surface area = (area of square) + 4 × (area of a triangular face)
= (10 × 10) + 4 ×
5 OX 9 ΔEBC
6 EX 10 ΔABE
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
T LM = 8 cm MN = 6 cm TO = 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
177Chapter 6 Surface Area and Volume
2 As the perpendicular height of the triangular face is not given, this must be calculated.
In the diagram, AM is the perpendicular height of the face.
In ΔAOM
= 64 + 9
= 73
AM =
=
=
= 138·5 cm2 (correct to 1 dec. pl.)
3 The perpendicular heights TA and TB must be calculated, as these are not given.
In ΔTOA, TA2 = AO2 + OT2
= 32 + 72
= 42 + 72
Now,
=
=
=
AO = 8 cm BC = 6 cm
B
------------------- ×+
In right rectangular pyramids, the opposite triangular faces are congruent.
LM MN×( ) 2 MN TB× 2
----------------------- 2 LM TA×
----------------------------- 2 8 58×× 2
Sam ple
pa ge
178 New Signpost Mathematics Enhanced 10 5.1–5.3
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 perpendicular height of each face and then calculate the surface area of 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 MS5·3·1 1 Calculate the area of each net.
a
2 Find the surface area of each square or rectangular pyramid. a
6
5
7·5 cm
9·5 cm
T
L M
O E
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
O B
BCDE is a rectangle. AO = 4 cm, ED = 10 cm, DC = 6 cm.
Sam ple
pa ge
179Chapter 6 Surface Area and Volume
Find the surface area of: a a square pyramid, base edge 6 cm, height 5 cm b 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 has a 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 faces b 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:
H
F
ABCD is a square. AB = 5 m, MO = 3 m, CG = 2 m
A
E
B
AO = OB = 9 cm CD = DE = FE = FC = 6 cm
4 cm
2 cm
3· 5
8
What fraction of a circle is each of the following sectors?
5 6 7
8 9 10 Evaluate πrs if r = 3·5 and s = 6·5. Answer correct to 1 dec. pl.
O O 120° O
O
180 New Signpost Mathematics Enhanced 10 5.1–5.3
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 the curved 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 1 Draw a semicircle of radius 10 cm. Step 2 Make a cone by joining opposite sides of the semicircle, as shown below. Step 3 Put the cone face down and trace the circular base. Measure the diameter of this base. Step 4 Calculate 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
sticky tape
A B
d BA
If you bring the two ‘B’s together, the sector
will bend to form a cone.
Sam ple
pa ge
181Chapter 6 Surface Area and Volume
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 =
=
= πrs Now, since the area of the sector = area of the curved surface,
curved surface area = πrs
Find the curved surface area of the following cones, giving answers in terms of π. a b
Worked examples 1 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.
Solutions 1 Surface area = πrs + πr2
= π × 5 × 8 + π × 52
= 40π + 25π = 65π cm2
(correct to 1 dec. pl.)
length of sector’s arc circumference of circle --------------------------------------------------------- area of circle×
2πr 2πs --------- πs2×
Surface area of a cone: surface area = πrs + πr2
where r = radius of the cone and s = slant height of the cone
Note: s h2 r2+=
s
r
h
2 First the slant height must be calculated. Now, s2 = 52 + 122
(Pythag. theorem) = 169
= 65π + 25π = 90π cm2
12
5
s
The height of a cone is the perpendicular height.
Exercise 6:03 Surface area of a cone MS5·3·1 1 For each cone shown, find the:
i radius ii height iii slant height a 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 cm b diameter = 16 cm; height = 15 cm
8 10
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 following cones, giving all answers in terms of π. a radius 8 cm and height 6 cm b radius 1·6 m and height 1·2 m c diameter 16 cm and height 15 cm d 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 cm c radius 12·5 cm and height 4·5 cm d diameter 1·2 m and height 60 cm e 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 the sector shown. In the cone that is formed, find: a the slant height b the radius c 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
height, not the vertical height, of the cone.
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A
CB
AB = 12 cm BO = 12 cm BC = 5 cm OD = 10 cm
9 cm
15 cm
6:04 Surface Area of Outcome MS5·3·1
a Sphere
Carry out the experiment outlined below to demonstrate that the reasoning is correct.
Step 1 Cut a solid rubber ball or an orange into two halves. The faces of the two hemispheres are identical circles.
Step 2 Push a long nail through the centre of a hemisphere, as shown.
Step 3 Using thick cord, cover the circular face of 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 4 Put 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 5 Turn 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...
1
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 the level 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 example A 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.
Solution Since the two hemispheres have the same radius, they will form a sphere if joined.
= πr3 + πr2h
= 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.
+=
196 New Signpost Mathematics Enhanced 10 5.1–5.3
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 metres b 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 in the diagram. Give the answer to the nearest cubic metre.
A house is to be built on a concrete slab which is 20 cm thick. The cross-sectional shape of the slab is shown in the diagram. Calculate the volume of concrete needed for the slab.
A steel tank is as shown in the diagram.
Given that the dimensions are external dimensions and that the steel plate is 2 cm thick, calculate the mass of the tank if the density of the steel is 7·8 g/cm3. Give the answer correct to one decimal place.
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4
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197Chapter 6 Surface Area and Volume
Calculate the volume of a concrete beam that has the cross-section shown in the diagram. The beam is 10 m long.
Calculate the mass of this beam if 1 m3 of concrete weighs 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 and the heights of the bottom cone, cylinder and top cone are 2·8 m, 8·5 m and 1·8 m respectively, calculate the volume of the tank correct to one decimal place.
A storage bin has been made from a square prism and a square pyramid. The top 1·8 m of the pyramid has been removed. Calculate the volume of the bin.
A glasshouse is in the shape of a square pyramid. Calculate the area of the four triangular faces to the nearest square metre if the side of the square is 20 m and the height of the pyramid is 17 m.
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11
198 New Signpost Mathematics Enhanced 10 5.1–5.3
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 from two truncated cones (frustrums). Calculate the volume of this bin.
composite solid • A solid that is formed
by joining simple solids.
joined by rectangular faces.
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
slant height
199Chapter 6 Surface Area and Volume
• 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 cm c a hemisphere of diameter 15 cm
Section 6:02
4·6 m
5·3 m
7·6 cm
4·3 cm
2·6 cm
12·3 cm
200 New Signpost Mathematics Enhanced 10 5.1–5.3
1 Calculate the volume and surface area of the pyramid shown.
2 A cone has a diameter of 16 cm and a height of 15 cm. Calculate the surface area of the cone in terms of π.
3 Calculate the surface area of a hemisphere that has a diameter of 16 cm. Give your answer correct to two significant figures.
4 A spherical shaped tank is to hold 100 m3. What radius to the nearest millimetre will give a volume closest to 100 m3?
5 Calculate the volume of the solid pictured. Give the answer correct to three significant figures.
6 Calculate the surface area of the solid shown. Give the answer correct to three significant figures.
7 A tank for holding liquid chemicals consists of a cylinder with two hemispherical ends as shown in the diagram. Calculate its volume (correct to 3 sig. figs.)
8 An octahedron is a double pyramid with all its edges equal in length. Calculate the surface area and volume of an octahedron with all its edges 6 cm in length.
Revision Chapter 6 Revision Assignment
9 cm
9 cm
6 cm
6 cm
• Engineers solve many surface area and volume problems in the design and construction of buildings.
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201Chapter 6 Surface Area and Volume
1 The value of a library is depreciated at a rate of 15% pa. If the library is presently valued at $800 000, what will its value be after 4 years?
2 ΔABC is equilateral with a side of length a units. AT, BS and RC are perpendicular heights of the triangle that meet at X. Find the lengths: a BX b XT
3 ABCD is a right triangular pyramid. Its base is an equilateral triangle of side units and the edges AB, AC and AD are all 6 units long. Find the height, AX and the volume of the pyramid.
4 Three positive whole numbers are multiplied in pairs. The answers obtained are 756, 1890 and 4410. What are the numbers?
5 The dot indicates the position of one chess queen on a chess board. How many more queens can you place on the board so that none of the queens threatens another?
6 One large sheet of paper was ruled up and folded. It was then cut along the fold shown in the diagram on the top to form an 8-page booklet. On the diagrams to the right, put the page number on each quarter (as has been done for 2 and 4).
Revision Chapter 6 Working Mathematically
A
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