1. Area and Volume Scale Factors
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Chapter 3
1.
Area and Volume Scale Factors
| Student Learning Objectives: |
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Why
Scale factors serve as a mathematical tool to understand how the area and volume of similar figures relate to each other. Whether you're working on a construction project or resizing a recipe, understanding these factors can be invaluable. The concept allows you to proportionally adjust the size of objects or spaces, making it easier to plan and execute various tasks. For example, if you're an architect, you can use scale factors to determine how the area of a room changes if its dimensions are altered. Similarly, in cooking, knowing how to adjust the volume of ingredients can save you from culinary disasters. Overall, mastering scale factors in the context of area and volume can offer practical solutions in diverse fields.
Lesson Settings & Tools
| | 13 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Area and Volume Scale Factors
Slide of 13
In this lesson, similar figures will be compared in terms of their surface areas and volumes.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Using Scale Factor of Similar Pyramids
On a trip to Egypt, Emily is visiting the Great Egyptian Museum. She sees two models of Pyramids, Khafra and Menkaure, on display. She decides to buy the model of Menkaure, but now as she exits the gift shop, she has become even more curious about the volume of Khafra.
Suppose that the models of the pyramids are similar. If the scale factor of the corresponding side lengths is 1:2 and the volume of the smaller pyramid is 20 cubic centimeters, what is the volume of the larger model?
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Discussion
Area Scale Factor
If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.
Let KLMN and PQRS be similar figures, and A_1 and A_2 be their respective areas. The length scale factor between corresponding side lengths is ab. Here, the following conditional statement holds true.
KLMN ~ PQRS ⇒ A_1/A_2 = (a/b )^2
Proof
The statement will be proven for similar rectangles, but this proof can be adapted for other similar figures.
The area of a rectangle is the product of its length and its width.
| Area of KLMN | Area of PQRS |
|---|---|
| A_1 = KL* LM | A_2 = PQ * QR |
By the definition of similar polygons, the corresponding side lengths are proportional and equal to the scale factor ab. KL/PQ= a/b [1.1em] LM/QR=a/b ⇔ KL = PQ * a/b [1.1em] LM = QR * a/b The next step is to substitute the expressions for KL and LM into the formula for A_1, which represents the area of KLMN.
A_1 = KL* LM
SubstituteII
KL= PQ * a/b, LM= QR * a/b
A_1 = ( PQ * a/b) ( QR * a/b)
▼
Simplify right-hand side
RemovePar
Remove parentheses
A_1 = PQ * a/b * QR * a/b
CommutativePropMult
Commutative Property of Multiplication
A_1 = a/b * a/b * PQ * QR
ProdToPowTwoFac
a* a=a^2
A_1 = (a/b )^2 * PQ * QR
AssociativePropMult
Associative Property of Multiplication
A_1 = (a/b )^2(PQ * QR)
Notice that the expression on the right-hand side is ( ab )^2 times the area of PQRS, or A_2.
A_1 = (a/b )^2( PQ* QR)
Substitute
PQ* QR= A_2
A_1 = (a/b )^2 A_2
DivEqn
.LHS /A_2.=.RHS /A_2.
A_1/A_2 = (a/b )^2
This proof has shown that the ratio of the areas of the similar rectangles is equal to the square of the ratio of their corresponding side lengths. This ratio is also called the area scale factor.
Scale Factor & & Area Scale Factor a/b & ⇒ & A_1/A_2 = (a/b )^2
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Example
Using Area Scale Factor to Determine an Unknown Area
Dominika is making a flyer for a concert and wants to compare the areas of the two pieces of paper. The widths of A4 and A2 papers are 21 and 42 centimeters, respectively.
If these two pieces of paper are similar and the area of A4 paper is about 624 square centimeters, find the area of the A2 paper. Round the answer to the nearest integer.
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Hint
If the scale factor of two similar figures is ab, then the ratio of their areas is a^2b^2.
Solution
The two pieces of paper are similar and two corresponding sides measure 21 centimeters and 42 centimeters.
Therefore, the scale factor is the ratio of these corresponding sides. Scale Factor: 21/42 = 1/2 Using this information, the ratio of the areas, or area scale factor, can be calculated by the theorem about the areas of similar figures. ccc Scale Factor & & Area Scale Factor [0.8em] 1/2 & ⇒ & 1^2/2^2= 1/4 Now, let A_2 be the area of the piece of A2 paper. Then, a proportion can be written using the area scale factor and the area of the A4 paper, which is 624 square centimeters.
624/A_2 = 1/4
▼
Solve for A_2
MultEqn
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
MultEqn
LHS * 4=RHS* 4
2496 = A_2
RearrangeEqn
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.
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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.
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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.
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
MultEqn
LHS * 2.5=RHS* 2.5
3/5 * 2.5 = x
MoveRightFacToNum
a/c* b = a* b/c
7.5/5 = x
CalcQuot
Calculate quotient
1.5 = x
RearrangeEqn
Rearrange equation
x = 1.5
The corresponding length in Figure A is 1.5 inches.
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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.
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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
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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.
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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
MultEqn
LHS * 90=RHS* 90
V_1 = 8/27 * 90
MoveRightFacToNum
a/c* b = a* b/c
V_1 = 720/27
CalcQuot
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.
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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.
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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
MultEqn
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
CalcQuot
Calculate quotient
(692.307692 ...) ( 1/10^(- 3) ) = R
FracToNegExponent
1/a^m=a^(- m)
(692.307692 ...) (10^3) = R
RearrangeEqn
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.
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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.
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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.
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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)
CalcQuot
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
MultEqn
LHS * 650=RHS* 650
0.82^2 * 650 = A_1
CalcPow
Calculate power
0.6724 * 650 = A_1
Multiply
Multiply
437.06 = A_1
RearrangeEqn
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.
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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.
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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
V_s ≈ 1299
The capacity of the smaller silo is about 1299 cubic meters.
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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.
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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
MultEqn
LHS * 8=RHS* 8
1 = 160/V_2
MultEqn
LHS * V_2=RHS* V_2
V_2 = 160
The volume of the larger model is 160 cubic centimeters.
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Area and Volume Scale Factors
Exercise 2.1
Ignacio's neighbor, Enrique, has bought a new Jaguar-print Jaguzzi
hot tub. Ignacio wants to one-up Enrique, so he will buy a Jaguzzi with twice the diameter and height of Enrique's Jaguzzi. Ignacio sneakily gathered the measurements from behind the fence.
How many times more water will Ignacio's Jaguzzi hold compared to Enrique's?
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It is a given that the Jaguzzi Ingacio wants to buy is twice the height and radius of Enrique's Jaguzzi. Therefore, the length scale factor is 1:2 or 12. Length Scale Factor= 1/2 The ratio of the volumes of similar solids is equal to the cube of the ratio of their corresponding linear measures. Let's put this information into action and write the volume scale factor for both hot tubs.
| Length Scale Factor | Volume Scale Factor |
|---|---|
| 1/2 | ( 1/2)^3=1/8 |
Since the volume of Enrique's hot tub is unknown, let V represent its volume. Assuming that the volume of Ignacio's hot tub is x times greater than Enrique's, we can write its volume as xV. Therefore, we can equate the volume scale factor to the ratio of V to xV. 1/8=V/xV By solving the equation for x, we will find how many times more water Ignacio's hot tub holds.
The Jaguzzi that Ignacio will buy can hold 8 times as much water as Enrique's Jaguzzi.
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E
Hint & Answer
S
Solution
Ramsha is reflooring her living room and has drawn the following sketch.
What is Ramsha's living room area in square meters if the length scale factor between the sketch and the room is 1:75? Round the answer to the nearest tenth.
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We are asked to find the area of the actual living room. Let's begin by calculating the area of the sketch.
Area of Sketch
Notice that we can divide the sketch into two rectangles.
We can then calculate the length of the small rectangle by subtracting 1.7 from 5
5-1.7= 3.3
By the same reasoning, we can find the height of the larger rectangle by subtracting 2.1 from 6.3.
6.3-2.1= 4.2
Let's update the labels of the rectangles to include the new measurements.
We can now find the area of the rectangles by using the formula for the area of a rectangle. Let's show the calculations in a table.
| Rectangle | A=wl | Area |
|---|---|---|
| Smaller | A= (3.3cm)(2.1cm) | A=6.93cm^2 |
| Larger | A= (5cm)(4.2cm) | A=21cm^2 |
By adding the areas of both rectangles, we will get the total area of the sketch. 6.93cm^2+21cm^2= 27.93 cm^2
Area of Living Room
We are told that the length scale factor is 1:75 or 175. Recall that the ratio of the areas of similar figures is equal to the square of the ratio of their corresponding side lengths. With this information, we can write the area scale factor for the sketch and the living room.
| Length Scale Factor | Area Scale Factor |
|---|---|
| 1/75 | (1/75)^2=1/75^2 |
We can now equate the area scale factor to the ratio of area of the sketch to area of the living room. Let A_1 be the area of the sketch and A_2 be the area of the living room. 1/75^2=A_1/A_2 Since we found that A_1 is 27.93 square centimeters, we can substitute this value into the expression and solve it for A_2. Let's do it!
The area of the living room is about 157 106 square centimeters.
Converting to Square Meters
Let's convert the area we found into square meters. Since 1 meter equals 100 centimeters, 1 square meter must equal 100 square centimeters. Therefore, we can write the following conversion factor. 1 m^2/100^2 cm^2 With this information, we can convert the area into square meters.
The area of the living room is about 15.7 square meters.
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How many times longer is the volume of the larger solid than the volume of the smaller solid?
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We will begin by finding the length scale factor between the solids. To do so, we can mark two pairs of corresponding sides of the figures. Let's do it!
We can use any pair of the measures of the corresponding sides to calculate the scale factor. In this case, we will use the sides whose measures are 3 and 6. Let's calculate the ratio between the measure of these sides to get the length scale factor. Length Scale Factor:3/6= 1/2 Recall that the ratio of of the volumes of similar solids is equal to the cube of the ratio of their corresponding linear measures. With this information, we can write the volume scale factor for the given solids.
| Length Scale Factor | Volume Scale Factor |
|---|---|
| 1/2 | ( 1/2)^3=1/8 |
The volume scale factor is 18 or 1:8. Therefore, the volume of the large solid is 8 times larger than the volume of the small solid.
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Area and Volume Scale Factors
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