Scale Factor, Area and Volume Explained

Alberta Grade 11 Mathematics

Measurement

Relating a length scale factor to area, surface area, and volume

A model, drawing, or object can be enlarged while keeping the same shape. If every length is multiplied by the same number, the areas and volumes do not change by that same number. This lesson explains how the length scale factor relates to area, surface area, and volume. The key idea is that area uses two dimensions, while volume uses three.

What you will learn

Explain what a scale factor tells you about corresponding lengths.
Use the square of a length scale factor to compare areas and surface areas.
Use the cube of a length scale factor to compare volumes.
Find a scale factor from area information and use it to compare volumes.

Review: lengths, area, and volume

A length measures one direction, such as the width of a sign. Area measures the amount of flat surface inside a shape. For a rectangle, area is length multiplied by width. Volume measures the space inside a three-dimensional object. For a rectangular prism, volume is length multiplied by width multiplied by height.

Units help show what is being measured. Length may be in centimetres, area in square centimetres, and volume in cubic centimetres. The powers in the units match the number of dimensions being measured.

A scale factor is the number that multiplies each original length to get its corresponding new length. A scale factor of

2

means every length doubles. A scale factor of

0.5

means every length becomes half as long. Corresponding lengths are lengths in matching positions on the original and enlarged or reduced shapes.

k=\frac{\text{new length}}{\text{original length}}

Length is one-dimensional; area is two-dimensional; volume is three-dimensional.
Use the same length scale factor for all corresponding lengths when the shapes keep the same shape.
A scale factor greater than $1$ enlarges; a scale factor between $0$ and $1$ reduces.

Why area uses the square of the scale factor

Start with a rectangle that is

4

cm wide and

7

cm long. Its area is

28

square centimetres. If every length is scaled by

1.5

, the new dimensions are

6

cm and

10.5

cm. The new area is

63

square centimetres. The area is multiplied by

63\div 28=2.25

, which is

1.5^2

.

This happens because area is found by multiplying two lengths. If each of those lengths is multiplied by

k

, then the product is multiplied by

k\times k

, or

k^2

. The same rule applies to surface area. Surface area is the total area of all the outside faces of a three-dimensional object. Each face has two dimensions, so each face area is scaled by

k^2

.

This rule works for enlargements and reductions. For example, if lengths are reduced by a factor of

0.5

, areas are multiplied by

0.5^2=0.25

. The new area is one quarter of the original area, not one half.

\text{area scale factor}=k^2

Area scale factor: square the length scale factor.
Surface area scale factor: square the length scale factor.
Keep area units square, such as $\text{cm}^2$ .

Why volume uses the cube of the scale factor

A rectangular prism has length, width, and height. Volume is found by multiplying all three dimensions. If each dimension is multiplied by

k

, the volume is multiplied by

k\times k\times k

, or

k^3

.

For example, if every length in a solid is doubled, its volume becomes

2^3=8

times as large. This does not mean that the solid is eight times taller. Each of its three dimensions doubles, and together those changes make eight times the volume.

Surface area and volume follow different rules, even for the same solid. A scale factor of

2

makes surface area four times as large, but volume eight times as large. Check which quantity the question asks about before choosing a power.

\text{volume scale factor}=k^3

Volume scale factor: cube the length scale factor.
For a reduction, cube the scale factor as well. A factor of $0.5$ gives $0.125$ of the original volume.
Keep volume units cubic, such as $\text{cm}^3$ .

Working backward from area

Sometimes the area scale factor is given, and the length scale factor is not. Since area changes by

k^2

, find the positive square root of the area scale factor to get

k

. A length scale factor must be positive because it compares actual lengths.

Once the length scale factor is known, use it to find the volume scale factor by cubing it. Do not cube the area scale factor directly. First connect area to length, then connect length to volume.

A good reasonableness check is to compare the powers. For an enlargement, the area and volume factors should both be greater than

1

. The volume factor should be the cube of the length factor, and the area factor should be its square.

k=\sqrt{\text{area scale factor}}

From an area scale factor, take the positive square root to find the length factor.
Then cube the length factor to find the volume factor.
Do not confuse a scale factor with the new measurement itself.

Worked example

Enlarging a rectangular sign

A rectangular sign is

4

cm by

7

cm. A similar sign is made with a length scale factor of

1.5

. Find the new dimensions and area, and compare the areas.

Sign dimensions before and after scaling

The original rectangle is $4$ cm by $7$ cm. Each length is scaled by $1.5$, so the new rectangle is $6$ cm by $10.5$ cm.

Scale each length
Multiply each original dimension by $1.5$ . This gives the dimensions of the new sign because the same scale factor applies to every corresponding length.
$4(1.5)=6, 7(1.5)=10.5$
Find each area
Area is length multiplied by width. Calculate the original and new areas, then compare the new area with the original.
$A_1=4(7)=28, A_2=6(10.5)=63$
Compare the areas
Divide the new area by the original area. The result matches the square of the length scale factor.
$\frac{63}{28}=2.25=1.5^2$

Answer: The new sign is

6

cm by

10.5

cm. Its area is

63\text{ cm}^2

, which is

2.25

times the original area of

28\text{ cm}^2

.

Check: The length factor is greater than

1

, so the new area should be larger. The area factor

2.25

is greater than

1

, as expected.

Worked example

Scaling a cube

A cube has side length

3

cm. A similar cube has side length

7.5

cm. Find the length scale factor, the surface-area scale factor, and the volume scale factor. Also find the new cube’s surface area and volume.

A face of the cube before and after scaling

A square face with side length $3$ cm scales by $2.5$ to side length $7.5$ cm. The face area factor is $2.5^2$; the full cube volume factor is $2.5^3$.

Find the length factor
Divide a new side length by the corresponding original side length. This gives the scale factor for every length of the similar cubes.
$k=\frac{7.5}{3}=2.5$
Find the surface-area factor
Surface area is made from face areas, and each face has two dimensions. Square the length scale factor.
$k^2=2.5^2=6.25$
Find the volume factor
Volume depends on three dimensions, so cube the length scale factor.
$k^3=2.5^3=15.625$
Calculate the new measurements
The original cube has six square faces and a volume equal to its side length cubed. Multiply each original measurement by its scale factor. SA_1=6(3^2)=54, SA_2=54(6.25)=337.5, V_2=3^3(15.625)=421.875

Answer: The length scale factor is

2.5

. The surface-area scale factor is

6.25

, and the volume scale factor is

15.625

. The new cube has surface area

337.5\text{ cm}^2

and volume

421.875\text{ cm}^3

.

Check: The length factor is greater than

1

. Both new measurements are larger than the original ones. Surface area uses a square factor, while volume uses a cube factor.

Worked example

Using area to compare volumes

A similar model has a surface area of

405\text{ cm}^2

. The original model’s surface area is

180\text{ cm}^2

, and its volume is

40\text{ cm}^3

. Find the length scale factor and the model’s new volume.

Surface-area ratio as a squared length factor

A representative rectangle scales by $1.5$ in each direction. Its area factor is $2.25$, matching the given ratio of total surface areas.

Find the area factor
Divide the new surface area by the original surface area. This ratio is the surface-area scale factor.
$\frac{405}{180}=2.25$
Recover the length factor
Surface area changes by the square of the length factor. Take the positive square root because lengths are positive.
$k=\sqrt{2.25}=1.5$
Find the volume factor and new volume
Volume changes by the cube of the length factor. Multiply the original volume by that factor to find the new volume.
$k^3=1.5^3=3.375, V_2=40(3.375)=135$

Answer: The length scale factor is

1.5

. The volume scale factor is

3.375

, so the new model’s volume is

135\text{ cm}^3

.

Check: The area factor is

2.25

, which is greater than

1

, so the length factor must be greater than

1

. The resulting volume is also greater than the original

40\text{ cm}^3

.

Common mistakes and how to avoid them

Multiplying area by the length scale factor instead of squaring it.

Correction: Area has two dimensions, so use the square of the length factor.

Using the square of the length factor for volume.

Correction: Volume has three dimensions, so cube the length factor.

Taking the square root of an area scale factor and choosing a negative value.

Correction: Use the positive square root. A scale factor compares positive lengths.

Giving an area in cubic units or volume in square units.

Correction: Use square units for area and surface area, and cubic units for volume.

Lesson summary

A length scale factor $k$ multiplies every corresponding length.
Area and surface area scale by $k^2$ .
Volume scales by $k^3$ .
From an area scale factor, take the positive square root to find the length factor before finding the volume factor.

Check your understanding

Question 1

A similar shape is enlarged by a length scale factor of

3

. By what factor does its area change?

$3$
$6$
$9$
$27$

Show answer and explanation

9

Area uses the square of the length scale factor:

3^2=9

.

Question 2

A solid’s lengths are reduced by a factor of

0.5

. What fraction of its original volume remains?

$0.5$
$0.25$
$0.125$
$0.75$

Show answer and explanation

0.125

Volume uses the cube of the length factor:

0.5^3=0.125

.

Question 3

The area scale factor between two similar objects is

16

. What is the length scale factor?

$4$
$8$
$16$
$256$

Show answer and explanation

4

The length factor is the positive square root of the area factor, so

\sqrt{16}=4

.

Key terms

Scale factor: The multiplier that changes an original length into its corresponding new length.
Similar shapes: Shapes with the same shape, where corresponding lengths have the same scale factor.
Surface area: The total area of all the outside faces or surfaces of a three-dimensional object.
Volume: The amount of space inside a three-dimensional object.
Corresponding lengths: Lengths in matching positions on the original and scaled shapes.

Continue through Math 20-2

View the complete Math 20-2 Alberta Grade 11 Mathematics curriculum and lessons

Tutor support for M3 · Relate scale factors to area, surface area, and volume

Working on M3 · Relate scale factors to area, surface area, and volume in Math 20-2? Bring the question and your own attempt. In a live session, you can show each step of your attempt and compare it with a worked example. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث M3 · ضریب مقیاس را به مساحت، مساحت سطح و حجم ربط دهید در درس Math 20-2، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید گام‌های تلاش خود را نشان دهید و آن را با یک نمونهٔ حل‌شده مقایسه کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation M3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

M3 · Scale Factor, Area and Volume Explained

Alberta Grade 11 Mathematics

Relating a length scale factor to area, surface area, and volume

What you will learn

Review: lengths, area, and volume

Why area uses the square of the scale factor

Why volume uses the cube of the scale factor

Working backward from area

Enlarging a rectangular sign

Scaling a cube

Using area to compare volumes

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 20-2

Tutor support for M3 · Relate scale factors to area, surface area, and volume

About this lesson and its review