DoAssignment study guide

E1.4 · Analyse how changing dimensions affects measures

Learn to analyse how changing dimensions affects measures through clear examples and targeted practice.

Ontario Grade 9 Mathematics

Geometry and Measurement

Ontario Grade 9 Mathematics — study topic E1.4

Imagine making a rectangular garden longer while keeping its width the same. The garden’s boundary gets longer, but its area also grows. These are different measures, so they do not always change in the same way. In this lesson, a dimension means a length used to describe a shape, such as its length, width, or height. A measure is a quantity we calculate from those dimensions, such as perimeter, area, or volume. We will compare the original and changed dimensions to see what happens to each measure.

What you will learn

Describe how a change in a shape’s dimensions affects its perimeter and area.
Describe how changing the dimensions of a box affects its volume.
Use calculations and comparisons to explain how a measure changes when one or more dimensions change.

1. Start with a rectangle

A rectangle has four sides. Opposite sides have the same length. Its perimeter is the distance around its edge. Its area is the amount of flat surface it covers. Perimeter is measured in units such as centimetres; area is measured in square units, such as square centimetres.

Suppose a rectangle is 5 cm long and 3 cm wide. Its perimeter is the total of its four sides:

5+3+5+3=16

cm. Its area is the number of square centimetres that cover it:

5\times3=15

cm².

Now keep the width at 3 cm and increase the length from 5 cm to 7 cm. The perimeter becomes 20 cm, an increase of 4 cm. The area becomes 21 cm², an increase of 6 cm². The two measures both increase, but by different amounts. This is why it matters to name the measure being discussed.

P=2l+2w, A=lw

Perimeter measures distance around a flat shape.
Area measures the surface inside a flat shape.
Changing a dimension can change more than one measure, and each measure may change differently.

2. Compare dimensions and measures

A useful way to analyse a change is to compare the original dimensions with the new dimensions. Calculate the measure both times, then describe how it changed. You can report the new measure, the difference, or how many times as large it became.

For example, if a rectangle’s length doubles while its width stays the same, its area doubles. Each unit of width now has twice as much length beside it. The perimeter does not generally double, because the unchanged width still contributes to the perimeter.

If both dimensions double, the result is different. The new area is four times the original area: there are twice as many rows and twice as many columns of equal-sized square units. The perimeter doubles because every side length is doubled.

The table uses a rectangle that is initially 5 units long and 3 units wide. Each change is compared with this same starting rectangle. A multiplier tells how many times the original measure the new measure is.

A=lw

Change one dimension at a time when you want to see its separate effect.
If both rectangle dimensions are multiplied by the same number, the perimeter is multiplied by that number.
If both rectangle dimensions are doubled, the area becomes four times as large.

3. Apply the same thinking to a box

A box has three dimensions: length, width, and height. Its volume is the amount of space inside it. Volume is measured in cubic units, such as cubic centimetres. For a rectangular box, multiply the three dimensions.

If only the length doubles while the width and height stay unchanged, the volume doubles. The box has twice as much length for the same width and height. If all three dimensions double, the volume becomes eight times as large: each of the three dimensions contributes a factor of 2.

A comparison helps explain the result. Doubling length and width makes the base four times as large. Doubling the height then stacks twice as many equal-sized layers. Together, these changes make the volume eight times as large.

For any dimension change, substitute the new dimensions into the volume rule and compare the result with the original volume. This method also works when dimensions change by different amounts. Be sure to use the same units for all dimensions before calculating.

V=lwh

Volume measures the space inside a three-dimensional object.
For a rectangular box, volume depends on length, width, and height.
Doubling all three dimensions makes the volume eight times the original volume.

4. A reliable comparison method

Use these steps when a question asks how a measure changes. First, identify the shape and the measure, such as perimeter, area, or volume. Next, record the original dimensions and the changed dimensions. Then calculate the measure in each case using the matching rule. Finally, compare the answers and state the change clearly.

A difference is found by subtracting the original measure from the new measure. A multiplier is found by dividing the new measure by the original measure. For example, a multiplier of 3 means the new measure is three times the original. Keep units with your answers: a difference in area uses square units, while a difference in volume uses cubic units.

Check whether your conclusion makes sense. If a dimension grows and the other dimensions remain positive and unchanged, area or volume should not become smaller. Also check that you did not confuse square units with cubic units. They describe different kinds of measures.

\text{multiplier}=\frac{\text{new measure}}{\text{original measure}}

Name the measure before choosing a calculation rule.
Compare the new result with the original result; do not rely only on the dimensions changing.
State whether the measure increased or decreased and by how much, or by what factor.

Effect of dimension changes on a 5-by-3 rectangle

Dimensions (length by width)	Perimeter	Area	Comparison with original area
5 by 3	16 units	15 square units	Original
10 by 3	26 units	30 square units	2 times as large
10 by 6	32 units	60 square units	4 times as large

Worked example

Changing the dimensions of a box

A rectangular box measures 4 cm by 3 cm by 2 cm. Its length changes from 4 cm to 8 cm, while its width and height stay the same. Analyse how its volume changes.

Identify the measure
The question asks about the space inside the box, so calculate volume. For a rectangular box, multiply its length, width, and height.
$V=lwh$
Find the original volume
Use the original dimensions. The result is in cubic centimetres because three lengths are multiplied.
$V=4\times3\times2=24\text{ cm}^3$
Find the new volume
Replace only the length with 8 cm. Keep the width and height unchanged, as the question states.
$V=8\times3\times2=48\text{ cm}^3$
Compare the volumes
Subtract to find the increase. Then compare the new volume with the original to describe the change as a multiplier.
$48-24=24, 48\div24=2$

Answer: The volume increases by 24 cm³, from 24 cm³ to 48 cm³. The new volume is twice the original volume.

Check: Only the length doubled, while the other dimensions stayed fixed. The volume therefore doubled, which agrees with the calculations.

Common mistakes and how to avoid them

Saying that doubling a rectangle’s length always doubles its perimeter.

Correction: Check all four sides. If only the length doubles, the unchanged width still contributes to the perimeter, so the perimeter does not generally double.

Saying that doubling both dimensions of a rectangle only doubles its area.

Correction: Both dimensions affect area. Doubling the length and the width makes the area four times as large.

Using square units for volume or cubic units for area.

Correction: Use square units for area and cubic units for volume. The units match the number of dimensions multiplied.

Comparing the new measure with the wrong starting value.

Correction: Write down the original measure first. Use it as the comparison point when finding a difference or multiplier.

Lesson summary

Perimeter measures the distance around a flat shape; area measures its covered surface; volume measures space inside a three-dimensional object.
A rectangle’s perimeter depends on its length and width, while its area is the product of those dimensions.
A rectangular box’s volume is the product of its length, width, and height.
Calculate the measure before and after a dimension change, then compare the results with correct units.

Check your understanding

Question 1

A rectangle is 6 cm by 2 cm. Its length doubles, while its width stays 2 cm. What happens to its area?

It doubles, from 12 cm² to 24 cm².
It triples, from 12 cm² to 36 cm².
It stays 12 cm².
It becomes four times as large, from 12 cm² to 48 cm².

Show answer and explanation

It doubles, from 12 cm² to 24 cm².

The original area is 6 × 2 = 12 cm². The new area is 12 × 2 = 24 cm², so it doubles.

Question 2

All three dimensions of a rectangular box are doubled. How does its volume compare with the original?

It is 2 times as large.
It is 4 times as large.
It is 6 times as large.
It is 8 times as large.

Show answer and explanation

It is 8 times as large.

Each of the three dimensions contributes a factor of 2. The volume multiplier is 2 × 2 × 2 = 8.

Question 3

A rectangle changes from 4 units by 3 units to 8 units by 6 units. What is the new area compared with the original area?

2 times as large
4 times as large
6 times as large
8 times as large

Show answer and explanation

4 times as large

The original area is 4 × 3 = 12 square units. The new area is 8 × 6 = 48 square units, and 48 ÷ 12 = 4.

Key terms

Dimension: A length used to describe a shape, such as its length, width, or height.
Perimeter: The total distance around a flat shape.
Area: The amount of flat surface covered by a shape.
Volume: The amount of space inside a three-dimensional object.
Multiplier: The number that tells how many times as large one measure is compared with another.

Continue through MTH1W

View the complete MTH1W Ontario Grade 9 Mathematics curriculum and lessons

Use the free MTH1W test and exam review guide

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation E1.4. 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