Use scale drawings to calculate lengths and areas

Ontario Grade 8 Mathematics

Spatial Sense

Read the scale, measure carefully, and connect the drawing to the real object.

A scale drawing shows an object in a different size while keeping its lengths in the same proportions. A map, a floor plan, or a sketch of a garden can be a scale drawing. The scale tells how a length on the drawing compares with the matching length in real life. For example, if

1\text{ cm}

on a plan represents

3\text{ m}

in a room, then a line measuring

4\text{ cm}

represents

12\text{ m}

. The drawing is smaller, but the comparison stays consistent. In this lesson, you will use scales to find lengths and areas. First, remember that perimeter measures the distance around a shape, while area measures the space inside it. Area is counted in square units, such as square metres.

What you will learn

Explain what a scale means in a drawing.
Use a scale to calculate an actual length or a drawing length.
Use scaled side lengths to calculate area.
Check that units and answers make sense.

1. Understand the scale

A scale is a statement that links a length on a drawing to a matching length in real life. In the scale

1\text{ cm}:4\text{ m}

, the colon means “is represented by.” Each

1\text{ cm}

on the drawing stands for

4\text{ m}

in real life. Keep the units attached to each part so you do not mix drawing lengths with actual lengths.

To find an actual length, count how many scale units fit along the drawing length, then multiply by the actual length represented by one scale unit. To find a drawing length, reverse the process: divide the actual length by the actual length represented by one drawing unit. Check that the answer is reasonable. An actual room should usually be longer than its small plan drawing.

\text{actual length}

=

\text{drawing length}

×actual length per drawing unit

Scale compares a drawing length with an actual length.
Multiply to go from drawing length to actual length.
Divide to go from actual length to drawing length.

2. Use a scale to calculate lengths

Suppose a path on a map measures

5\text{ cm}

and the map scale is

1\text{ cm}:2\text{ km}

. Each centimetre represents two kilometres. Five centimetres represent five groups of two kilometres, so the path is

10\text{ km}

long.

Sometimes the unknown is the length to draw. If a real wall is

18\text{ m}

long and the plan scale is

1\text{ cm}:3\text{ m}

, the drawing length is

18\div3=6\text{ cm}

. Division is useful here because you are finding how many groups of

3\text{ m}

fit into

18\text{ m}

.

Before calculating, make sure the measured drawing length and the scale use the same drawing unit. For example, if the scale uses centimetres but the drawing length is measured in millimetres, convert one of them first. Remember:

10\text{ mm}=1\text{ cm}

.

\text{drawing length}

=

\text{actual length}

\divactual length per drawing unit

Multiply by the scale amount when finding actual length.
Divide by the scale amount when finding drawing length.
Use matching units and include the correct unit in the answer.

3. Use scale drawings to calculate areas

Area is the number of square units that cover a flat shape without gaps or overlaps. For a rectangle, multiply its length by its width. If both side lengths are scaled by the same amount, the area changes by that amount in both directions. This is why the area scale factor is applied twice.

For example, if every drawing length represents

5

times that length in real life, a drawing rectangle that is

2\text{ cm}

by

3\text{ cm}

represents a real rectangle that is

10\text{ cm}

by

15\text{ cm}

. The drawing area is

6\text{ cm}^2

, while the actual area is

150\text{ cm}^2

. The actual area is

25

times the drawing area because

5\times5=25

.

A reliable method is to use the scale to find the actual side lengths first, then calculate the actual area. This also helps you keep the units clear. If actual side lengths are in metres, the area is in square metres, written

\text{m}^2

.

A_{\text{rectangle}}=\text{length}\times\text{width}

Find the actual side lengths before finding actual area.
For a rectangle, area is length multiplied by width.
Area units are squared, such as $\text{m}^2$ or $\text{cm}^2$ .

4. A careful checking routine

Read the scale and identify which length it describes on the drawing and in real life. Decide whether the question asks for an actual length, a drawing length, or an area. Then choose multiplication or division for a length. For an area, find the actual side lengths and use the shape's area rule.

Finally, check the units and size of your result. A length answer should use a length unit, such as centimetres or metres. An area answer should use square units. If the result is smaller than expected, check whether you divided when you should have multiplied, or whether you used a drawing length instead of an actual length.

Label drawing lengths and actual lengths clearly.
Do not report an area in ordinary length units.
Estimate whether the result is a sensible size.

Worked example

Find an actual path length

A park map uses the scale

1\text{ cm}:250\text{ m}

. A path measures

6.4\text{ cm}

on the map. What is its actual length?

Identify the scale
Each centimetre on the map represents $250\text{ m}$ in the park.
$1\text{ cm}:250\text{ m}$
Multiply by the map length
There are $6.4$ map centimetres, so multiply $6.4$ by the actual distance represented by one centimetre.
$6.4\times250=1600$
State the unit
The scale gives actual distances in metres, so the path is $1600\text{ m}$ long.
$1600\text{ m}=1.6\text{ km}$

Answer: The actual path is

1600\text{ m}

, or

1.6\text{ km}

.

Check: The actual path should be much longer than its map length. The scale gives hundreds of metres for each centimetre, so the result is reasonable.

Worked example

Find a length to draw

A rectangular room is

7.2\text{ m}

long. A floor plan uses the scale

1\text{ cm}:1.2\text{ m}

. How long should the room be on the plan?

Read the actual length per drawing unit
Each $1\text{ cm}$ on the plan stands for $1.2\text{ m}$ in the room.
$1\text{ cm}:1.2\text{ m}$
Divide to find drawing length
Find how many groups of $1.2\text{ m}$ fit into the room's $7.2\text{ m}$ length.
$7.2\div1.2=6$
Attach the drawing unit
The quotient counts centimetres on the plan, because each group of $1.2\text{ m}$ represents one plan centimetre.
$6\text{ cm}$

Answer: The room should be drawn

6\text{ cm}

long.

Check: The room length is six times the actual length represented by one plan centimetre, so the plan length is six centimetres.

Worked example

Calculate the actual area

A rectangular garden is shown on a scale drawing as

3\text{ cm}

by

2\text{ cm}

. The scale is

1\text{ cm}:4\text{ m}

. Find the actual area of the garden.

Rectangular garden on the drawing

Coordinates show drawing lengths in centimetres. The rectangle is 3 cm by 2 cm; the stated scale converts its sides to actual lengths.

Find each actual side length
Each drawing centimetre represents $4\text{ m}$ . Multiply each side of the drawing by $4$ to get the matching garden side.
$3\times4=12\text{ m},\quad 2\times4=8\text{ m}$
Use the rectangle area rule
A rectangle's area is its length multiplied by its width. Use the actual side lengths, not the drawing lengths.
$12\times8=96$
Write square units
Both actual side lengths are in metres, so the area is in square metres.
$96\text{ m}^2$

Answer: The actual garden area is

96\text{ m}^2

.

Check: The drawing area is

6\text{ cm}^2

. Each side is enlarged by a factor of

4

, so the area is enlarged by

4\times4=16

. Since

6\times16=96

, the result agrees.

Common mistakes and how to avoid them

Multiplying when the question asks for a drawing length.

Correction: When the actual length is known, divide by the actual length represented by one drawing unit.

Using drawing side lengths to report an actual area.

Correction: First convert each side to its actual length, then calculate the area.

Writing an area in metres instead of square metres.

Correction: Area uses square units. If both side lengths are in metres, write the area in

\text{m}^2

.

Applying the length scale only once to an area.

Correction: Both dimensions change. For a rectangle, convert both side lengths before multiplying.

Lesson summary

A scale links a drawing length to an actual length.
Multiply drawing length by the scale amount to find actual length.
Divide actual length by the scale amount to find drawing length.
For area, convert the actual side lengths first, then use the area rule and square units.

Check your understanding

Question 1

A drawing uses the scale

1\text{ cm}:5\text{ m}

. A fence is

4\text{ cm}

long on the drawing. How long is it in real life?

$9\text{ m}$
$20\text{ m}$
$1.25\text{ m}$
correctIndex: 1,

Show answer and explanation

20\text{ m}

Multiply the drawing length by the actual length represented by one centimetre:

4\times5=20\text{ m}

.

Question 2

A real path is

15\text{ m}

long. At a scale of

1\text{ cm}:3\text{ m}

, how long should it be drawn?

$45\text{ cm}$
$12\text{ cm}$
$5\text{ cm}$
correctIndex: 2,

Show answer and explanation

5\text{ cm}

Divide the actual length by the actual length represented by one drawing centimetre:

15\div3=5\text{ cm}

.

Question 3

A rectangle on a scale drawing is

2\text{ cm}

by

3\text{ cm}

. The scale is

1\text{ cm}:2\text{ m}

. What is its actual area?

$12\text{ m}^2$
$24\text{ m}^2$
$10\text{ m}^2$
correctIndex: 1,

Show answer and explanation

24\text{ m}^2

The actual side lengths are

4\text{ m}

and

6\text{ m}

. Their product is

24\text{ m}^2

.

Key terms

Scale: A comparison between a drawing length and the matching actual length.
Scale drawing: A drawing that keeps lengths in the same proportions as the object or place.
Area: The amount of flat space inside a shape, measured in square units.
Square unit: A unit used to measure area, such as a square metre or square centimetre.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation E1.3. 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

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E1.3 · Use scale drawings to calculate lengths and areas

Ontario Grade 8 Mathematics

Read the scale, measure carefully, and connect the drawing to the real object.

What you will learn

1. Understand the scale

2. Use a scale to calculate lengths

3. Use scale drawings to calculate areas

4. A careful checking routine

Find an actual path length

Find a length to draw

Calculate the actual area

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review