Proportions with Perimeter, Area,

and VolumeChapter 11.5

ObjectiveTo discover the relationship

between the perimeters, areas and volumes of similar

figures

[You will need graph paper]

On Graph Paper• Draw rectangle ABCD with length and width of 16

and 12

• Draw rectangle EFGH with length and width of 12 and 9

• Write a similarity statement for the two rectangles

Yours should look something like these

H

G

FE

D C

BA

12

16

9

12

ABCD~EFGH

Ratios

Compare the larger rectangle to the smaller rectangle.

Write the ratio of any two corresponding sides.

HG

FE

D C

BA

12

16

9

12

3

4

12

163

4

9

12

Remember, this is called the LINEAR

RatioAKA: Similarity RatioScale Factor

Ratios

Calculate the perimeter of each rectangleWhat is the ratio of the larger perimeter to the

smaller?

HG

FE

D C

BA

12

16

9

12

3

4

42

56

P=56 P=42

Ratios

Since the perimeter is a LINEAR measurement, it is in the same LINEAR ratio.

HG

FE

D C

BA

12

16

9

12

P=56 P=42

Linear Similarity ConjectureThe ratios of any corresponding linear measures of similar figures are equal to the ratio of corresponding sides

Linear Measurements

Linear Measurements• Perimeter• Length• Width• Height• Diameter

• Radius• Circumference

Now back to the rectangles…

Linear ratio MUST be in simplest form

AreaCalculate the Area of each rectangle

What is the ratio of the larger to smaller areas?

HG

FE

D C

BA

12

16

9

12

A=192 A=1089

16

108

192

AreaHow does this ratio compare to the linear ratio?(The linear ratio was )

HG

FE

D C

BA

12

16

9

12

A=192 A=1089

16

108

192

3

4

AreaDraw another set of similar rectangles on your

paper and see if your theory works again

(try starting with a 5x7 rectangle and choosing a scale factor to make a second rectangle)

AreaProportional Areas Conjecture

If corresponding sides of two similar polygons or the

radii of two circles compare in the ratio , then

their areas compare in the ratio

n

m

2

2

2

n

mor

n

m

Linear ratio MUST be in simplest form

Check This Out (?)NCTM Applet for Perimeter and Area

Area ExamplesThe ratio of the corresponding midsegments of two

similar trapezoids is 4:5. What is the ratio of their areas?

25

16

5

42

2

Linear (L):

5

4Area (A):

Area Examples

2

5

4

10

L:

Find the Linear and Area ratios

A:

4

25

2

52

2

Area ExamplesWhat is the area of circle N (in terms of π)?

PN

A=560π cm

mq

4

1

q

m

Fun Fact: ALL circles are similar!

16

1

4

12

2

L: A:

235

5601656016

1

cmx

x

x

VolumeConsider these rectangular prismsAre all of their corresponding linear measures

proportional?What is the linear ratio?What is the area ratio?

1

323

1.5

4.5

2

34

9

2

32

2

L: A:

VolumeFind the volume of each prismWhat is the ratio of the volumes?

V=20.25 V=6

8

27

6

25.20

1

323

1.5

4.5

Volume

and the ratio of VOLUMES is

Is there a relationship?8

272

3L: A:

4

9

V=20.25 V=61

323

1.5

4.5

VolumeProportional Volumes Conjecture

If corresponding edges (or radii, height, etc.) of two

similar solids compare in the ratio , then their

areas compare in the ratio

n

m

3

3

3

n

mor

n

m

Check This Out (?)NCTM Applet for Volume of Similar Solids

Volume ExamplesThe corresponding heights of two similar cylinders

is 2:5. What are the Linear, Area and Volume ratios?

125

8

5

23

3

2 5 5

2L: A:

25

4V:

Volume ExamplesTriangular prisms X and Y are similarThe linear ratio is ¾. Find the area and volume ratios.

9ft k

VX = 35.1ft3

X Y

64

27

4

33

3

4

3L: A:

16

9

V:

Volume ExamplesTriangular prisms X and Y are similarThe linear ratio is ¾. Find K

9ft k

VX = 35.1ft3

X Yftk

k

k

75.6

27494

3

64

274

3

16

9V:L: A:

Volume ExamplesTriangular prisms X and Y are similarThe linear ratio is ¾. Find volume of prism Y

9ft k

VX = 35.1ft3

X Y28.14

7.947)(641.3564

27

ftV

V

V

Y

Y

Y

64

274

3

16

9V:L: A:

The Ratios

Linear

Area Volumen

m

2

2

n

m3

3

n

mYou can’t jump

between area and volume without

going through linear