Unlocking Area and Volume with Scale Factors: A Guide to Proportional Measures

LHS * A_2=RHS* A_2

624/A_2 * A_2 = 1/4 * A_2

FracMultDenomToNumber

a/A_2* A_2 = a

624 = 1/4 * A_2

MoveRightFacToNumOne

1/b* a = a/b

624 = A_2/4

LHS * 4=RHS* 4

2496 = A_2

Rearrange equation

A_2 = 2496

The area of the A2 paper is 2496 square centimeters. Dominika figures that is just the right amount of area to promote the rock band Twenty-One Scale Breakers.

Example

Using Area Scale Factor to Determine an Unknown Side

It was just learned that if the length scale factor of two similar figures and one of the areas of the figures are known, then the unknown area can be found. Next, consider if both areas but only the side length of one figure are known. It will be possible to solve for the other similar figure's corresponding side length.

The diagram shows two similar figures. Figure A has an area of 9 square inches, and Figure B has an area of 25 square inches.

If a side length of Figure B is 2.5 inches, find the length of the corresponding side in the other shape.

Hint

If the scale factor of two similar figures is ab, then the ratio of their areas is a^2b^2.

Solution

Recall that the ratio of areas of two similar figures is equal to the square of the ratio of their corresponding side lengths. ccc Scale Factor & & Area Scale Factor [0.8em] a/b & ⇒ & (a/b)^2 Since the areas are given, the following proportion can be written. (a/b)^2 = 9/25 Now, take square roots of both sides of the equation to find the value of ab, the scale factor. Keep in mind that only the principal roots will be considered because only positive numbers make sense in this situation.

(a/b)^2 =9/25

▼

Solve for a/b

SqrtEqn

sqrt(LHS)=sqrt(RHS)

a/b=sqrt(9/25)

SqrtQuot

sqrt(a/b)=sqrt(a)/sqrt(b)

a/b=3/5

FracToScale

\dfrac a b = a:b

a:b=3:5

The scale factor of the figures is 3:5. Finally, with the scale factor and knowing that the side length of Figure B is 2.5 inches, the length of the corresponding side in Figure A represented by x can be found.

3/5=x/2.5

▼

Solve for x

LHS * 2.5=RHS* 2.5

3/5 * 2.5 = x

MoveRightFacToNum

a/c* b = a* b/c

7.5/5 = x

Calculate quotient

1.5 = x

Rearrange equation

x = 1.5

The corresponding length in Figure A is 1.5 inches.

Pop Quiz

Practice Finding Linear Scale Factor Given Areas

Determine the linear scale factor of the shape on the right to the shape on the left. If necessary, round the answer to two decimal places.

Discussion

Volume Scale Factor

For similar three-dimensional figures, the volume scale factor and the length scale factor are also related.

If two figures are similar, then the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths.

Let Solid A and Solid B be similar solids and V_1 and V_2 be their respective volumes. The length scale factor between corresponding linear measures is ab. Given these characteristics, the following conditional statement holds true.

SolidA ~ SolidB ⇒ V_1/V_2 = (a/b)^3

Proof

The statement will be proven for similar rectangular prisms, but this proof can be adapted to prove other similar solids. As shown in the diagram, let a_1, a_2, and a_3 be the dimensions of Solid A and b_1, b_2, and b_3 be the dimensions of Solid B.

The volume of a rectangular prism is the product of its base area and its height.

Volume of Solid A	Volume of Solid B
V_1 = a_1* a_2 * a_3	V_2 = b_1 * b_2 * b_3

By the definition of similar solids, the side lengths are proportional and equal to the scale factor ab. a_1/b_1=a/b [1.1em] a_2/b_2=a/b [1.1em] a_3/b_3=a/b ⇔ a_1 = b_1 * a/b [1.1em] a_2 = b_2 * a/b [1.1em] a_3 = b_3 * a/b The next step is to substitute the expressions for a_1, a_2, and a_3 into the formula for V_1, the volume of Solid A.

V_1 = a_1 * a_2 * a_3

SubstituteExpressions

Substitute expressions

V_1 = ( b_1 * a/b) ( b_2 * a/b) ( b_3 * a/b)

▼

Simplify right-hand side

RemovePar

Remove parentheses

V_1 = b_1 * a/b * b_2 * a/b * b_3 * a/b

CommutativePropMult

Commutative Property of Multiplication

V_1 = a/b * a/b * a/b * b_1 * b_2 * b_3

ProdToPowThreeFac

a* a* a=a^3

V_1 = (a/b )^3 * b_1 * b_2 * b_3

AssociativePropMult

Associative Property of Multiplication

V_1 = (a/b )^3 (b_1 * b_2 * b_3)

Notice that the expression on the right-hand side is ( ab )^3 times the volume of Solid B.

V_1 = (a/b )^3 (b_1 * b_2 * b_3)

Substitute

b_1 * b_2 * b_3= V_2

V_1 = (a/b )^3 V_2

DivEqn

.LHS /V_2.=.RHS /V_2.

V_1/V_2 = (a/b )^3

As shown, the ratio of the volumes of the similar prisms is equal to the cube of the ratio of their corresponding linear measures. This ratio is also called the volume scale factor.

Scale Factor & & Volume Scale Factor a/b & ⇒ & V_1/V_2 = (a/b )^3

Example

Using Volume Scale Factor to Determine an Unknown Volume

The scale factor of two similar figures can be used to find the volume of one of the figures when the volume of the other figure is known.

The suitcase company Case-O-La produces snazzy suitcases of various sizes. When a large-sized suitcase is bought, the company offers its cabin-sized version at a discounted price to the same customer.

The large-sized suitcase has a height of 27 inches and a volume of 90 liters. If the cabin-sized suitcase has a height of 18 inches, determine its volume. Round the answer to one decimal place.

Hint

If the scale factor of two similar figures is ab, then the ratio of their volumes is ( ab)^3.

Solution

The suitcases can be considered as two similar rectangular prisms with heights 27 and 18 inches.

Similar solids have the same shape and all of their corresponding sides are proportional. The ratio of the corresponding linear dimensions of the similar solids is the scale factor. Height of small suitcase/Height of big suitcase= 18/27 If the scale factor of two similar solids is a:b, then the ratio of their corresponding volumes is a^3:b^3. Now, raise the scale factor to the third power to obtain the ratio of the volumes.

a/b=18/27

ReduceFrac

a/b=.a /9./.b /9.

a/b=2/3

RaiseEqn

LHS^3=RHS^3

(a/b)^3=(2/3)^3

PowQuot

(a/b)^m=a^m/b^m

a^3/b^3=2^3/3^3

CalcPow

Calculate power

a^3/b^3=8/27

FracToScale

\dfrac a b = a:b

a^3:b^3=8:27

The ratio of the volumes is 8:27. Now, let V_1 be the volume of the small suitcase. Since the volume of the big suitcase is 90 liters, the ratio of V_1 to 90 is 8:27.

V_1/90=8/27

▼

Solve for V_1

LHS * 90=RHS* 90

V_1 = 8/27 * 90

MoveRightFacToNum

a/c* b = a* b/c

V_1 = 720/27

Calculate quotient

V_1 = 26.6666666...

RoundDec

Round to 1 decimal place(s)

V_1 ≈ 26.7

The volume of the small suitcase is about 26.7 liters.

Example

Using Volume Scale Factor to Determine an Unknown Side

After reading a physics magazine, Mark feels confident in estimating the radius of the Sun. To do so, he will use the volumes of the Sun and Earth, which are 1.41 * 10^(18) and 1.08 * 10^(12) cubic kilometers, respectively.

If the radius of the Earth is about 6300 kilometers, help Mark find the radius of the Sun.

Hint

Sun and Earth can be regarded as two similar spheres. Therefore, the volume scale factor can be used to find the radius of the Sun.

Solution

The Sun and Earth are two similar spheres. Consequently, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear measures, which in this case is the ratio of their radii. Let r and R be the radii of the Earth and Sun, respectively. ccc Scale Factor & & Ratio of the Volumes [0.8em] r/R & ⇒ & (r/R)^3 Given that the volumes are known, the volume scale factor can be used to find the scale factor of the Earth to the Sun. (r/R)^3 = 1.08 * 10^(12)/1.41 * 10^(18) Next, take the cube roots of both sides of the equation to find the value of rR, the length scale factor.

(r/R)^3 = 1.08 * 10^(12)/1.41 * 10^(18)

▼

Solve for a/b

WriteProdFrac

Write as a product of fractions

(r/R)^3 = (1.08/1.41)( 10^(12)/10^(18))

DivPow

a^m/a^n= a^(m-n)

(r/R)^3= (1.08/1.41)(10^(- 6))

RadicalEqn

sqrt(LHS)=sqrt(RHS)

r/R = sqrt((1.08/1.41)(10^(- 6)))

RootProd

sqrt(a* b)=sqrt(a)*sqrt(b)

r/R= sqrt(1.08/1.41) * sqrt(10^(- 6))

UseCalc

Use a calculator

r/R = (0.914958...)( 10^(- 2))

RoundDec

Round to 2 decimal place(s)

r/R ≈ (0.91)( 10^(- 2))

Write in scientific notation

r/R ≈ 9.1 * 10^(- 3)

Finally, with the scale factor and knowing that the radius of the Earth is about 6300 kilometers, the radius of the Sun R can be found. The ratio of 6300 to R is equal to the scale factor.

6300/R = 9.1 * 10^(- 3)

▼

Solve for R

LHS * R=RHS* R

6300 = (9.1 * 10^(- 3))(R)

DivEqn

.LHS /(9.1 * 10^(- 3)).=.RHS /(9.1 * 10^(- 3)).

6300/9.1 * 10^(- 3) =R

WriteProdFrac

Write as a product of fractions

(6300/9.1) ( 1/10^(- 3) ) = R

Calculate quotient

(692.307692 ...) ( 1/10^(- 3) ) = R

FracToNegExponent

1/a^m=a^(- m)

(692.307692 ...) (10^3) = R

Rearrange equation

R = (692.307692 ...) (10^3)

RoundInt

Round to nearest integer

R ≈ (692)(10^3)

Write in scientific notation

R ≈ 6.92 * 10^5

The radius of Sun is about 6.92 * 10^5, or 692 000, kilometers.

Pop Quiz

Practice Finding Linear Scale Factor Given Volumes

The applet shows the volumes of two similar solids. Determine the linear scale factor of the blue solid to the orange solid. If necessary, round to two decimal places.

Two similar solids varying between prisms, cylinders, pyramids, cones and spheres.

Example

Finding Area Given Volume of Similar Solids

The corresponding faces of two similar three-dimensional figures are also similar. Subsequently, the ratio of the areas of the corresponding faces is proportional to the square of the length scale factor of the figures.

Dylan has a golden retriever and a chihuahua. He buys two similar doghouses, whose corresponding side lengths are proportional. When he is about to finish painting the doghouses, he realizes that there is not enough paint for the front face of the small doghouse!

Dylan knows the volumes of each doghouse. They are about 63 500 and 34 500 cubic inches. He also knows that the front face of the big doghouse has an area of 650 square inches. Help Dylan determine the area of the front face of the small doghouse. This will help him determine how much more paint to buy. Round the answer to the nearest integer.

Hint

Start by finding the length scale factor. Then, use the ratio for the areas of similar figures.

Solution

The length scale factor will be found first. To do so, the cube root of the volume scale factor will be calculated. Recall that the ratio of volumes of two similar solids is equal to the cube of the ratio of their corresponding side lengths. ccc Scale Factor & & Volume Scale Factor [0.8em] a/b & ⇒ & (a/b)^3 A proportion can be written using the ratio of the volumes. (a/b)^3 = 34 500/63 500 Now, take cube roots of both sides of the equation to find the value of ab, the length scale factor.

(a/b)^3 = 34 500/63 500

▼

Solve for a/b

RadicalEqn

sqrt(LHS)=sqrt(RHS)

a/b = sqrt(34 500/63 500)

Calculate quotient

a/b = sqrt(0.543307...)

UseCalc

Use a calculator

a/b = 0.815984 ...

RoundDec

Round to 2 decimal place(s)

a/b ≈ 0.82

The area scale factor can be found by squaring the scale factor for length. ccc Scale Factor & & Area Scale Factor [0.8em] 0.82 & ⇒ & 0.82^2 Finally, knowing that the larger doghouse's front face area is about 650 square inches, the corresponding area in the other one can be calculated. Let A_1 be that area. Then, 0.82^2 should be equal to the ratio of A_1 to 650.

0.82^2 =A_1/650

▼

Solve for A_1

LHS * 650=RHS* 650

0.82^2 * 650 = A_1

CalcPow

Calculate power

0.6724 * 650 = A_1

Multiply

437.06 = A_1

Rearrange equation

A_1 = 437.06

RoundInt

Round to nearest integer

A_1 ≈ 437

The smaller doghouse has a front face with an area about 437 square inches. He can now go shopping for more paint to appease his cool chihuahua.

Since the corresponding faces of two similar three-dimensional figures are similar, it can be concluded that the ratio of the surface areas is the square of the scale factor.

Example

Finding Volume of Similar Composite Solids

A three-dimensional figure is called a composite solid if it is the combination of two or more solids. Like similar solids, if the corresponding linear measures of two composite solids are proportional, the composite solids are said to be similar. Therefore, their length scale factor can be determined and used to find certain characteristics of the shapes.

The amount of material used to construct the larger silo is three times that of the smaller one. That is, the surface area of the larger silo is three times as large as the surface area of the smaller silo. These two silos can be considered to be similar solids. Furthermore, each silo is composed of a cone and a cylinder as shown.

If the volume of the larger silo is 6750 cubic meters, find the volume of the smaller silo. Round the answer to the nearest integer.

Hint

Use the area scale factor to determine the length scale factor.

Solution

The given silos are similar composite solids. Since they are similar, their corresponding linear measures are proportional. Therefore, each silo can be considered as a whole. To find the volume of the smaller silo, these steps will be followed.

The ratio of their surface areas will be used to determine the length scale factor.
The length scale factor will be used to determine the volume scale factor.
Finally, the volume scale factor will be used to determine the volume of the smaller silo.

It is given that the surface area of the larger silo is three times as large as the surface area of the smaller silo.

To find the length scale factor, consider its relationship to the surface areas. Recall that for areas of similar figures, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Refer to the smaller silo's side length as a and the larger silo's side length as b. ccc Length Scale Factor & & Area Scale Factor [0.8em] a/b & ⇒ & (a/b)^2 Since the surface area of the larger silo is three times the surface area of the smaller silo, the ratio of the surface area, small to large, is 1:3. With that information, the following equation can be expressed. (a/b)^2 = 1/3 Next, this equation can be simplified to solve for the length scale factor. Begin by taking the square root of both sides of the equation.

(a/b)^2 = 1/3

▼

Solve for a/b

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt((a/b)^2) = sqrt(1/3)

SqrtPowToNumber

sqrt(a^2)=a

a/b = sqrt(1/3)

SqrtQuot

sqrt(a/b)=sqrt(a)/sqrt(b)

a/b = sqrt(1)/sqrt(3)

CalcRoot

Calculate root

a/b = 1/sqrt(3)

Now, the length scale factor can be used to find the volume scale factor. To do so, the length scale factor needs to be raised to the third power. ccc Length Scale Factor & & Volume Scale Factor [0.8em] a/b=1/sqrt(3) & ⇒ & (a/b)^3 = (1/sqrt(3))^3 Finally, knowing that the volume of the larger silo is about 6750 cubic meters, the volume of the smaller silo V_s can be calculated. Similar to the areas, the ratio of the volumes should be equal to the volume scale factor.

V_s/6750 = (1/sqrt(3))^3

▼

Solve for V_s

PowQuot

(a/b)^m=a^m/b^m

V_s/6750 = 1^3/(sqrt(3))^3

BaseOne

1^a=1

V_s/6750 = 1/(sqrt(3))^3

LHS * 6750=RHS* 6750

V_s = 6750/(sqrt(3))^3

UseCalc

Use a calculator

V_s = 1299.038105 ...

RoundInt

Round to nearest integer

V_s ≈ 1299

The capacity of the smaller silo is about 1299 cubic meters.

The similarity makes it possible to solve for certain characteristics of a wide range of shapes, like prisms, spheres, composite solids, and pyramids.

Closure

Relationship Between Length, Area, and Volume Scale Factor

In this course, the relationships between the length scale factor, area scale factor and volume scale factor have been discussed. If the scale factor between two similar figures is ab, then the ratio for their areas and volumes can be expressed as the table shows.

Length Scale Factor	Area Scale Factor	Volume Scale Factor
a/b	( a/b )^2	( a/b )^3

Considering these expressions, the challenge presented at the beginning can be solved with more confidence.

Emily knows that the models in the museum are similar pyramids and the scale factor between the corresponding side lengths is 1:2.

If the volume of the smaller model is 20 cubic centimeters, find the volume of the larger model.

Hint

If the scale factor of two similar figures is ab, then the ratio of their volumes is ( ab)^3.

Answer

To find the volume of the larger model, first, find the volume scale factor. Note that the volume scale factor is equal to the cube of the length scale factor. The length scale factor is given as 1:2, or 12. Length Scale Factor & & Volume Scale Factor 1/2 & ⇒ & (1/2)^3 = 1/8 An equation can now be written using the known volume scale factor and the ratio of the model's volumes. Recall that it is given that the volume of the smaller model is 20 cubic centimeters. Substitute that value into the equation to solve for the volume of the larger model.

1/8 = V_1/V_2

Substitute

V_1= 20

1/8 = 20/V_2

LHS * 8=RHS* 8

1 = 160/V_2