year 10 surface area and volumedobmaths.weebly.com/uploads/8/9/9/8/8998387/year... · chapter 10...

Year 10 Mathematics Surface Area and Volume Practice Test 4

Name__________________________

1 Find the perimeter of each of the following shapes. (Where appropriate, state your

answer correct to 2 decimal places)

C h a p t e r 1 0 S u r f a c e a r e a a n d v o l u m e

393

Maths Quest 10 NSW 5.2 P/ Final Pages / 27/1/06

Perimeter

The perimeter of a plane figure is the total length of the outside boundary of that figure.For some figures, a formula can be used to calculate the perimeter. In others, particu-larly irregular figures, lengths of all the sides need to be added. Certain lengths mayneed to be calculated first using Pythagoras’ theorem or trigonometry.

Since perimeter is a length, the units used will be millimetres, centimetres, metres orkilometres.

The table below shows some regular figures and the corresponding formula forfinding the perimeter.

All other shapes have their perimeter found by adding the side lengths.

Shape

Square Rectangle Circle

Formula for calculating perimeter

P

=

4

l

, where

l

is the side length

P

=

2(

l

+

b

), where

l

is the length and

b

is the breadth or width

P

=

Circumference

C

=

2

!

r

, where

r

is the radiusAs 2

r

=

d

(diameter),

C

=

!

d

l

l

br

Find the perimeter of each of the following shapes. (Where appropriate, state your answer correct to 2 decimal places.)a b c

THINK WRITEa The shown shape is a parallelogram. Add the

side lengths.a P = 5 + 12 + 5 + 12 = 34 cm

b The given shape is a circle, whose radius is given. Write the formula for the circumference that contains the radius.

b C = 2!r

Identify the value of r. r = 7

Substitute the values of r and ! into the formula and use a calculator to evaluate, correct to 2 decimal places.

C = 2 " ! " 7 = 43.98 cm

c The shape is a regular hexagon. There are 6 sides of equal length, so the perimeter is 6 times the side length.

c P = 6 " 5 = 30 cm

12 cm

5 cm

7 cm 5 cm

1

2

3

1WORKEDExample

MQ 10 NSW 5.2 P - 10_tb Page 393 Friday, January 27, 2006 9:10 PM

2 Find the perimeter of each of the following shapes, correct to 2 decimal places

a)

394 M a t h s Q u e s t 1 0 f o r N e w S o u t h W a l e s 5 . 2 P a t h w a y


Irregular shapes will need to have calculations made to calculate the length of aparticular side.

Application of perimeter is seen in many real problems.

Find the perimeter of each of the following shapes, correct to 2 decimal places.a b

THINK WRITEa The perimeter consists of three

straight sides and a semicircle with diameter 6 cm. Find the circumference of a circle with diameter 6 and halve it. (Use the formula that contains diameter.)

a Curved length = C

= (!d) = (! " 6)

= " 18.84 = 9.42 cm

Add the curved length to the lengths of the three straight sides.

P = 9.42 + 8 + 6 + 8 = 31.42 cm

b The perimeter consists of two straight sides and an arc. The length

of the arc is of the circumference

of a circle with radius 7 cm.

b Curved length = " C

= " 2!r

= " 2 " ! " 7

= 14.66 cmAdd the arc length and the length of the straight sides.

P = 14.66 + 7 + 7 = 28.66 cm

8 cm

6 cm 7 cm 120°

1 12---

12---

12---

12---

2

1

120360---------

120360---------

120360---------

120360---------

2

2WORKEDExample

A rectangular paddock 70 m by 48 m needs to be fenced with four rows of wire. What is the total length of wire required to complete the fencing?

THINK WRITESince the paddock is rectangular in shape, write the formula for the perimeter of a rectangle.

P = 2(l + b)

Identify the values of the pronumerals. l = 70, b = 48Substitute the values of the pronumerals into the formula to find the perimeter of the paddock.

P = 2(70 + 48) = 2 " 118 = 236 m

Since four rows of wire are required, to find the total length multiply the perimeter of the paddock by 4.

Length of wire required = 4 " P = 4 " 236 = 944 m

1

2

3

4

3WORKEDExample


b)



Irregular shapes will need to have calculations made to calculate the length of aparticular side.

Application of perimeter is seen in many real problems.

Find the perimeter of each of the following shapes, correct to 2 decimal places.a b

THINK WRITEa The perimeter consists of three

straight sides and a semicircle with diameter 6 cm. Find the circumference of a circle with diameter 6 and halve it. (Use the formula that contains diameter.)

a Curved length = C

= (!d) = (! " 6)

= " 18.84 = 9.42 cm

Add the curved length to the lengths of the three straight sides.

P = 9.42 + 8 + 6 + 8 = 31.42 cm

b The perimeter consists of two straight sides and an arc. The length

of the arc is of the circumference

of a circle with radius 7 cm.

b Curved length = " C

= " 2!r

= " 2 " ! " 7

= 14.66 cmAdd the arc length and the length of the straight sides.

P = 14.66 + 7 + 7 = 28.66 cm

8 cm

6 cm 7 cm 120°

1 12---

12---

12---

12---

2

1

120360---------

120360---------

120360---------

120360---------

2

2WORKEDExample

A rectangular paddock 70 m by 48 m needs to be fenced with four rows of wire. What is the total length of wire required to complete the fencing?

THINK WRITESince the paddock is rectangular in shape, write the formula for the perimeter of a rectangle.

P = 2(l + b)

Identify the values of the pronumerals. l = 70, b = 48Substitute the values of the pronumerals into the formula to find the perimeter of the paddock.

P = 2(70 + 48) = 2 " 118 = 236 m

Since four rows of wire are required, to find the total length multiply the perimeter of the paddock by 4.

Length of wire required = 4 " P = 4 " 236 = 944 m

1

2

3

4

3WORKEDExample


3 A rectangular paddock 70 m by 48 m needs to be fenced with four rows of wire. What is the total length of wire required to complete the fencing?

4 The length of a rectangular lawn is three times its breadth or width. If the perimeter of the lawn is 56 m, find its dimensions (length and breadth).

5 Find the area of these shapes a)

C h a p t e r 1 0 S u r f a c e a r e a a n d v o l u m e 403


Area

Where appropriate in this exercise, give answers correct to 2 decimal places.1 Find the areas of the following shapes.

2 Find the area of the following ellipses. a b

remember1. Area measures the amount of surface within the boundaries of the figure.2. The units for measuring area are mm2, cm2, m2 and km2.3. Land area is usually measured in hectares (ha) where 1 ha = 10 000 (or 104) m2.4. Areas can be calculated by using formulas that are specific to the given plane

figure.5. Areas of composite figures can be calculated by adding the areas of the simple

figures making the composite figure or by calculating the area of an extended figure and subtracting the extra area covered.

remember

10B10.2

Area ofsquares,

rectangles,triangles

and circlesMathcad

Area

GC program– TI

Measurement

GC program– Casio

Measurement

EXCEL Spreadsheet

Perimeterand area

7 m6 m18 cm

7 mm

8 mm 13 mm

4 cm4 cm

12 cm

8 cm

12 cm

18 cm

15 cm

a b c

d e f

g h i10 cm

25 cm

15 cm

10 cm

WORKEDExample

5a9 mm

4 mm

12 mm

5 mm


b)



Area

Where appropriate in this exercise, give answers correct to 2 decimal places.1 Find the areas of the following shapes.

2 Find the area of the following ellipses. a b

remember1. Area measures the amount of surface within the boundaries of the figure.2. The units for measuring area are mm2, cm2, m2 and km2.3. Land area is usually measured in hectares (ha) where 1 ha = 10 000 (or 104) m2.4. Areas can be calculated by using formulas that are specific to the given plane

figure.5. Areas of composite figures can be calculated by adding the areas of the simple

figures making the composite figure or by calculating the area of an extended figure and subtracting the extra area covered.

remember

10B10.2

Area ofsquares,

rectangles,triangles

and circlesMathcad

Area

GC program– TI

Measurement

GC program– Casio

Measurement

EXCEL Spreadsheet

Perimeterand area

7 m6 m18 cm

7 mm

8 mm 13 mm

4 cm4 cm

12 cm

8 cm

12 cm

18 cm

15 cm

a b c

d e f

g h i10 cm

25 cm

15 cm

10 cm

WORKEDExample

5a9 mm

4 mm

12 mm

5 mm


6 Find the Surface Area of this triangular prism



The surface area can be found using the formula generated below.

SA = Afront + Aback + Atop + Abottom + Aleft side + Aright side

SA = (l ! h) + (l ! h) + (l ! b) + (l ! b) + (b ! h) + (b ! h)

= lh + lh + lb + lb + bh + bh

= 2lh + 2lb + 2bh

= 2(lh + lb + bh)

The surface area of a rectangular prism (or a cuboid) of length l, breadth b and height h is given by the formula SA = 2(lh + lb + bh).A special case of the rectangular prism is the cube where all sides are equal (l = b = h).

Cube:SA = 6 l2

l

Find the surface area (SA) of this triangular prism.

THINK WRITEThe ends of the triangular prism (the front and back) are congruent triangles. Write the formula for the area of a triangle.

Atriangle = bh

Identify the value of the pronumerals. b = 6, h = 4Substitute the values of the pronumerals and calculate the area of each end.

Afront = Aback = ! 6 ! 4 = 12 cm2

The other faces are rectangles. Write the formula for the area of a rectangle.

Arectangle = lb

Identify the values of the pronumerals. For Aleft side and Aright side : l = 20, b = 5For Abottom : l = 20, b = 6

Substitute the values of the pronumerals into the formula and so calculate the area of each rectangular face.

Aleft side = Aright side = 20 ! 5

= 100 cm2

Abottom = 6 ! 20 = 120 cm2

To find the surface area, add areas of all faces together. Remember to include the appropriate unit.

SA = (2 ! 12) + (2 ! 100) + 120 = 344 cm2

4 cm6 cm

5 cm

5 cm

20 cm

1 12---

2

3 12---

4

5

6

7

7WORKEDExample


7 Find the Surface Area of this cylinder to 1 decimal place



Surface area of a cylinderThe top and bottom of a cylinder are congruent circles while the curved surface ofthe cylinder if it were cut open and unfolded would form a rectangle. The net of thecylinder is shown below.

If the cylinder has a radius of r and a height of h as shown in the net above then itsSA can be calculated as follows.

Radius of circles = rLength of the rectangle = circumference of circular ends

= 2!rWidth of rectangle = h

SA = Atop + Abottom + Aside

= !r2 + !r2 + lb = 2!r2 + 2!rh

The surface area of a cylinder is given by the formula: SA = 2!r2 + 2!rh, where r is the radius of the base and h is the height of a cylinder.

r

h

h

2 r!

A = r 2!

a Use the formula A = 2!rh to find the area of the curved surface of the cylinder, correct to 1 decimal place.

b Use the formula SA = 2!r2 + 2!rh to find the surface area of the cylinder, correct to 1 decimal place.

THINK WRITE

a Write the formula for the area of the curved surface of a cylinder.

a A = 2!rh

Identify the values of the pronumerals. r = 2, h = 3Substitute the values of the pronumerals into the formula.

A = 2 " ! " 2 " 3

Evaluate and round to 1 decimal place. A = 37.7 m2 b Write the formula for the surface area

of a cylinder.b SA = 2!r2 + 2!rh

Identify the values of the pronumerals (these are the same as in part a).

r = 2, h = 3

Substitute the values of the pronumerals into the formula.

SA = 2 " ! " 22 + 2 " ! " 2 " 3

Evaluate correct to 1 decimal place and include the appropriate unit.

= 62.8 m2

3 m

2 m

1

2

3

4

1

2

3

4

8WORKEDExample


8 Find the surface area of a sphere of radius 5 cm (correct to 1 decimal place)

9 Find the surface area of the solid shown below.



Surface area of composite solidsThe surfaces of some solids are combinations of quadrilaterals and triangles. To findthe surface area of these solids, find the area of each face and add the areas together.Look for pairs or groups of faces that have the same area.

Find the surface area of the solid shown below.

THINK WRITE

The solid has 4 rectangular faces.

Calculate the area of each rectangular face using the formula Arectangle = lb.

Top face: l = 15, b = 12 Atop = 15 ! 12 = 180 cm2

Right side: l = 17, b = 12 Aright side = 17 ! 12 = 204 cm2

Bottom face: l = 30, b = 12 Abottom = 30 ! 12 = 360 cm2

Left side: l = 12, b = 8 Aleft side = 12 ! 8 = 96 cm2

The other 2 faces (front and back) are identical trapeziums. Write the formula for the area of a trapezium.

Atrapezium = (a + b) ! h

Identify the values of the pronumerals.

a = 15, b = 30, h = 8

Substitute the values of the pronumerals into the formula and evaluate the area of one of these faces.

Afront = Aback = ! (15 + 30) ! 8

= ! (45) ! 8

= 180 cm2

Add all of the areas together to find the surface area of the solid. Remember to include the correct units.

SA = Atop + Aright side + Abottom + Aleft side + Afront + Aback

= 180 + 204 + 360 + 96 + 2 ! 180 = 1200 cm2

15 cm

17 cm12 cm

8 cm

30 cm

1

2

3 12---

4

5 12---

12---

6

11WORKEDExample


10 Find the volume of the following solids a)

b)

11 Find the volume of the sphere of radius 9 cm. Answer correct to 1 decimal place.

12 Find the volume of the following solids a)

b)

13 Calculate the volume of the composite solid shown



Volume of composite figuresTo calculate the volume of a composite solid, the solid is divided into components,individual volumes of the components calculated and then added. This technique isshown in the following worked example.

Find the volume of each of the following solids.a b

THINK WRITE

a Write the formula for finding the volume of a cone.

a V = !r2h

Identify the values of r and h. r = 8, h = 10Substitute and evaluate. V = " ! " 82 " 10

= 670.2 cm3

b Write the formula for volume of a pyramid.

b V = Ah

Find the area of the square base. A = l2 where l = 8A = 82

= 64 cm2

Identify the value of h. h = 12Substitute and evaluate. V = " 64 " 12

= 256 cm3

8 cm

10 cm 12 cm

8 cm

1 13---

2

3 13---

1 13---

2

3

4 13---

17WORKEDExample

Calculate the volume of the composite solid shown.

THINK WRITEThe given solid is a composite figure, made up of a cube and a square-based pyramid.

V = Volume of cube + Volume of pyramid1

18WORKEDExample

1.5 m

3 m

Continued over page




Note that the triangular prism in the above worked example was not standing on itscross-section (the triangle). As a result, the dimension that we usually call ‘height’ didnot really represent a height of the prism in a true sense of the word, but rather itsdepth. Thus, it is important to emphasise here that what is referred to as ‘height’ of aprism simply means the dimension perpendicular to its cross-section. If a prism standson its cross-section, the ‘height’ will indeed represent the physical height of a prism (asin part a of the above worked example). Otherwise it will represent the prism’s depthor length (as in part b).

Volume of spheres Volume of a sphere of radius, r, can be calculated using the formula: V = !r3.

Find the volumes of the following solids.a b

THINK WRITEa Write the formula for the volume of a cylinder. a V = !r2h

Identify the value of the pronumerals. r = 14, h = 20Substitute and evaluate. V = ! " 142 " 20

#12 315.0 cm3

b Write the formula for the volume of a triangular prism.

b V = bh " H

Identify the value of the pronumerals. (Note that h is the height of the triangle and H is the depth of the prism.)

b = 4, h = 5, H = 10

Substitute and evaluate. V = " 4 " 5 " 10

= 100 cm3

14 cm

20 cm

10 cm4 cm

5 cm

123

1 12---

2

3 12---

15WORKEDExample

43---

Find the volume of the sphere of radius 9 cm. Answer correct to 1 decimal place.

THINK WRITE

Write the formula for the volume of a sphere.

V = !r3


Substitute and evaluate. V = " ! " 93

= 3053.6 cm3

1 43---

2

3 43---

16WORKEDExample




Note that the triangular prism in the above worked example was not standing on itscross-section (the triangle). As a result, the dimension that we usually call ‘height’ didnot really represent a height of the prism in a true sense of the word, but rather itsdepth. Thus, it is important to emphasise here that what is referred to as ‘height’ of aprism simply means the dimension perpendicular to its cross-section. If a prism standson its cross-section, the ‘height’ will indeed represent the physical height of a prism (asin part a of the above worked example). Otherwise it will represent the prism’s depthor length (as in part b).

Volume of spheres Volume of a sphere of radius, r, can be calculated using the formula: V = !r3.

Find the volumes of the following solids.a b

THINK WRITEa Write the formula for the volume of a cylinder. a V = !r2h

Identify the value of the pronumerals. r = 14, h = 20Substitute and evaluate. V = ! " 142 " 20

#12 315.0 cm3

b Write the formula for the volume of a triangular prism.

b V = bh " H

Identify the value of the pronumerals. (Note that h is the height of the triangle and H is the depth of the prism.)

b = 4, h = 5, H = 10

Substitute and evaluate. V = " 4 " 5 " 10

= 100 cm3

14 cm

20 cm

10 cm4 cm

5 cm

123

1 12---

2

3 12---

15WORKEDExample

43---

Find the volume of the sphere of radius 9 cm. Answer correct to 1 decimal place.

THINK WRITE

Write the formula for the volume of a sphere.

V = !r3


Substitute and evaluate. V = " ! " 93

= 3053.6 cm3

1 43---

2

3 43---

16WORKEDExample






THINK WRITE


a V = !r2h


= 670.2 cm3


b V = Ah


= 64 cm2


= 256 cm3

8 cm

10 cm 12 cm

8 cm

1 13---

2

3 13---

1 13---

2

3

4 13---

17WORKEDExample




18WORKEDExample

1.5 m

3 m

Continued over page






THINK WRITE


a V = !r2h


= 670.2 cm3


b V = Ah


= 64 cm2


= 256 cm3

8 cm

10 cm 12 cm

8 cm

1 13---

2

3 13---

1 13---

2

3

4 13---

17WORKEDExample




18WORKEDExample

1.5 m

3 m

Continued over page


year 10 surface area and volumedobmaths.weebly.com/uploads/8/9/9/8/8998387/year... · chapter 10...

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