Scale Factor and the relationship to area and volume

GLE 0706.2.30706.4.3

SPI: 0706.2.70706.4.3

Remember these rules !!

If 2 shapes are similar, it is by

multiplication

The number that we multiply by is the scale factor.

Scale factor is the ratio of corresponding parts.

Let’s review, what we know about similar shapes.

2.8 cm

x

5.6 cm

1.75 cm

2.8 1.75=x5.6

9.8 = 2.8 x2.82.8

x = 3.5cm

Try again

6 m

x

1.5 m

8 m

6 8=x1.5

12 = 6 x66

x = 2m

Remember: if shapes are similar it is because they are related by multiplication.

If a shape doubles, the scale factor is 2; if the shape triples in size, the scale factor is 3, and so on.

1.75 cm

3.5 cm

2.8 cm

5.6 cm

The second quadrilateral is twice as tall and twice as wide.

This means the scale factor is 2.

Scale factor and Area

1.75 cm

3.5 cm

2.8 cm

5.6 cm

The rule: to find the area of the second shape multiply the area of the first times the scale factor squared.

The scale factor is 2, so its square is 4…let’s test it.

1.75 cm

3.5 cm

2.8 cm

5.6 cm

Since the scale factor is 2, the shape is twice as tall and twice as wide.

(l * w) scale factor2

1.75 x 2.8 x 22

1.75 x 2.8 x 4 = 19.6 cm 2

Check yourself.. is that = to 5.6 x 3.5 ?

Remember, you are using the scale factor to find area when you don’t know the length of all the sides.

The second tripled in size, so the scale factor is 3

45

6

15

The area of the second:Area of first x

10 x 32 10 x 9 = 90

Is this true if you use the formula ½ b x h

2

Yes, ½ 15 * 12 = 90

The reason this works is because area is increased by length and width. If both dimensions are increased, you are square – ing.

6m

18m

2m The second shape is 3x as big, so the scale factor is 3.

The area of the first times theScale factor 2.12 x 32 =

12 x 9 = 108m2

10 cmIf the scale factor is ½ what would be the area of the smaller?

Area of the first times the scale factor2

10 * 10 * (1/2)2=

100 * ¼ = 25 cm2

Scale factor and volume

Now let’s look at scale factor and volume.

The rule is to multiply the volume of the known times the scale factor3.

Remember you are increasing the length, width, and height of a shape, thus cube - ing.

10 cm3 cm

5 cm

30 cmThe scale factor is 3, because the size has tripled. What is the volume of the larger prism?

Hint: volume of small x scale factor3

150 x 33= 150 x 27 = 4050 cm3

10 m5 m

5 m30 mWhat is the scale factor?

Hint: volume of small x scale factor3

750 x (1/2)3= 750 x 1/8 = 93.75 cm3

½ the shape is reduced by 2

What is the volume of the small?

Now Check yourself !!

length width Scale factor

15 10 2

3 5 3

20 40 1/2

Answers:

600

135

200

Now Check yourself !!

Answers:

28818432288

length width height Scale factor

3 6 2 2

6 12 4 4

30 60 20 1/5

Assume all the shapes are cubes