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Foundations of Math 11 Section 1.4 – Perimeter, Area and Surface Area of Similar Figures ♦ 35

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1.4 Perimeter, Area and Surface Area of Similar Figures

The squares shown below are similar. The corresponding sides are in a 1: 2 ratio. What is the ratio of

the perimeter?

1

1

11

2

2

22

Perimeter of larger squarePerimeter of smaller square

= 2 + 2 + 2 + 21+1+1+1

= 84= 2

1

Let us look at a similar triangle in a ratio of 2 :1 .

a

b

ch

2a

2b

2c2h

Perimeter of larger trianglePerimeter of smaller triangle

= 2a + 2b+ 2ca + b+ c

=2 a + b+ c( )

a + b+ c= 2

1

Thus, we can state the following:

If the scale factor of two similar figures is a : b , then the ratio of the perimeters is a : b .

Mt. Douglas Secondary

36 ♦ Chapter 1 – Rates and Scale Factor Foundations of Math 11


What about the area of similar figures? Let us look at the two squares again.

1

1

11

2

2

22

Area of larger squareArea of smaller square

= 2 × 21×1

= 41

Many would guess 2 :1 , but the area is 4 :1 .

Let us look at the similar triangle. Remember, the area of a triangle is 12

the base times the height: A = 1

2bh .

a

b

ch

2a

2b

2c2h

Perimeter of larger trianglePerimeter of smaller triangle

=12 2b( ) 2h( )

12 bh

= 2bh12 bh

= 212

= 41

Thus, we can state the following:

If the scale factor of two similar figures is a : b , then the ratio of the areas is a2 : b2 .

Two triangles are similar if all 3 angles in each triangle are equal. In fact, if two angles from each triangle are equal, the third angle must also be equal (3 angles of a triangle add up to 180°).

When writing the statement that the two triangles are similar, the order of the letters should indicate which angles are equal.

A

B C

F

ED

Given ∠A = ∠F and ∠B = ∠E

ΔABC ≈ ΔFED

(or ΔBCA ≈ ΔEDF

or ΔCAB ≈ ΔDEF)

The statement ΔCBA ≈ ΔDEF implies

1. ∠C = ∠D

2. ∠B = ∠E

3. ∠A = ∠F

4. CB ≈ DE

5. BA ≈ EF

6. CA ≈ DF




Example 1 ABCD is a parallelogram.

A

E

FB

CD

3 6

9 Find the following ratios:

a)

area of ΔAEFarea of ΔBCF

b)

area of ΔAEFarea of ΔDEF

Solution:▼

a) ΔAEF ~ ΔBCF because all 3 angles of ΔAEF are equal to angles of ΔBCFAFFB

= 36= 1

2thus area of ΔAEF

area of ΔBCF= 12

22 = 14

b) ΔAEF ~ ΔDECAFDC

= 39= 1

3thus area of ΔAEF

area of ΔDEF= 12

32 = 19

Example 2 A plane parallel to the base of a cone divides the cone into two pieces.

3 cm

5 cm

Find the ratio of a) the surface area of the small cone to the surface area of the larger cone.

b) the lateral area of the small cone to the lateral area of the bottom piece (the bottom piece is called the frustum).

Solution:▼ a) height of small cone

height of larger cone= 3

3+ 5= 3

8, thus surface area = 32

82 = 964

b) The lateral area of a cone does not include the base, only the sides.

Thus, lateral surface area =

32

82 – 32 = 964 – 9

= 955




Example 3 The two cylinders are similar. If the surface area of the largest cylinder is 54 cm3 ,

find the surface area of the smaller cylinder.

2 3

Solution:▼ Since the ratio of the radius is 2 :3 , the surface area of the cylinder is 22 :32 or 4 :9 .

49= x

54

9x = 4× 54

x = 4 × 549

x = 24

Surface area of small cylinder is 24 cm3 .

Example 4 Given the trapezoid ABCD with diagonals AC and BD, find

a) the ratio of areas I to III, and

b) the ratio of areas of I to II.

A B

E

CD

I

IIIII

3

5

Solution:▼

a) ΔABE ~ ΔCDE Thus, area Iarea III

= 32

52 = 925

b) ΔABE is not similar to ΔADE butarea of ΔABE = 1

2 h ⋅EBarea of ΔADE = 1

2 h ⋅DE= EB

DE

where h is the vertical height

from A to DB

A

D E B

h

EB and DE are corresponding sides of ΔABE and ΔCDE. Thus, area I

area II= 3

5




1.4 Exercise Set

1. The table refers to similar figures. Complete the table.

Scale Factor 1 :3 2:3 3:4 6:9

Perimeter Ratio

3 :5 6:4 5:3

Area Ratio

1 :4 9:16 36:25

2. a) Two cones have radii 8 and 12. The heights

are 14 and 21. Are the cones similar?

b) The heights of two pyramids are 12 and 16. The bases are square with sides 8 and 10. Are the pyramids similar?

3. If two circles have radii 8 and 12, what is the

ratio of the circumferences? Of the areas?

4. If the area of two circles are 16 and 36 ,

what is ratio of the radii? Of the

circumferences?




5. a) Three circles with centres O, P and Q are tangent to b) each other. The radius of the circles are 1, 2 and 3.

Find the ratio of

perimeter ΔOPQcircumference of circles (O + P +Q)

.

a) 1 : 1 b) 1 : 2 c) 1 : 3 d) 1 : e) : 1

Arc AB = 16

of the circumference

of the circle.

Find the ratio of

length of ABdiameter of circle O

.

a) : 2 b) : 1 c) 2 : 1 d) 2 2 : 1 e) 1 : 2

c) Circle O is rolled one complete revolution along a

flat surface.

Find the ratio of

distance rolleddiameter of circle

.

a) 1 : 3 b) 1 : 2 c) : 1 d) 2 : 1 e) 1 :

d) Find the ratio of circumference of Pcircumfernce of O

.

a) 1 : 4 b) 4 : 1 c) 1 : 2 d) 2 : 1 e) 3 : 1

P

O

O

A B




6. ABCD is a parallelogram. Find the ratio.

A B F

C

E

D

8 4

a)

Area of ΔAFDArea of ΔBFE

=

b)

Area of ΔBFEArea of ΔCDE

=

7. ABCD is a parallelogram. Find the ratio of the areas for each pair of triangles.

EA

B

G

CD

F

2 3

a) ΔEAF and ΔEBC

b) ΔEAF and ΔCDF

c) ΔEBG and ΔCDG




8. Consider the triangles I, II and III.

I II III

3 5 6

a) Find the ratio of the areas of triangles I

and II.

b) Find the ratio of the areas of triangles I

and III.

c) Find the ratio of the areas of triangles II

and III.

9. The figure is a trapezoid.

IIV II

III

2

3

a) Find the ratio of the areas of triangles I

and III.

b) Find the ratio of the areas of triangles I

and II.

c) Find the ratio of the areas of triangles I

and IV.

d) Find the ratio of the areas of triangles II

and IV.




10. Given ΔABC with D the mid-point of AC and

E the mid-point of AB.

(Hint: 2EF = FC, 2DF = FB)

A

D

CB

E

F

If the area of ΔEDF = 1 , find the area of

a) ΔBFC

b) ΔEBF

c) ΔCDF

d) ΔAED

11. The area of a parallelogram ABCD is 30 cm2 .

A

D C

B

F

E

If AE = 2EB, find the area of

a) ΔADE

b) ΔEBC

c) ΔEBF

d) ΔDCF




12. Find the ratio of the areas of figure I and II.

2

3

I

II

13. A rectangle of length twice the width is

inscribed in an isosceles triangle. Find the

ratio of the areas of regions I and II.

I

II

14. If the radius of a sphere is doubled, how much

larger is the surface area? If tripled, how

much larger?

S = 4 r 2( )

15. The scale model of a Lamborghini sports car is

1: 200 . If it requires 15 litres of paint for the

real car, how much paint is needed for the

scale model in mL? (1 litre = 1000 mL)




3 cm

6 cm

I

II

16. A parallel plane divides

a cone into two regions.

Find the ratio of the

lateral surface area of

region I to region II.

17. Two similar cylinders have lateral areas of

36 and 64 . Find the ratios of their

a) heights

b) total surface areas

18. A hemispherical room requires 4 cans of paint

to paint the floor. How many cans are

required to paint the walls?

19. If the surface area of a sphere is quadrupled,

how much larger is the radius?




20. Two spheres have radius x and y. What is the

ratio of their surface areas?

21. A sphere fits exactly

into a cube. What is

the ratio of the surface

area of the sphere to

the surface area of

the cube?


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