What Is a Scale Drawing?

A scale drawing is a drawing that represents a real object at a different size while keeping all measurements in the same proportion.

A scale drawing may be:

• Smaller than the real object.
• Larger than the real object.
• Occasionally the same size.

Scale drawings are useful when the actual object is too large or too small to draw conveniently.

Examples include:

• Maps.
• Building plans.
• Blueprints.
• Model vehicles.
• Floor plans.
• Engineering diagrams.

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Scale

The scale tells us how measurements on a drawing relate to measurements in real life.

For example:

1 cm represents 5 m

means that every:

1 cm

on the drawing represents:

5 m

in the real object.

If a wall measures 4 cm on the drawing, its actual length is:

4 × 5 = 20 m

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Why Scale Drawings Are Useful

Imagine trying to draw a school building at its actual size.

That would obviously be impractical.

Instead, we reduce all of its measurements by the same proportion.

Likewise, a very small object can be enlarged so its details are easier to see.

The important idea is that the object's proportions remain unchanged.

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Scale Factors

A scale factor tells us how much a figure has been enlarged or reduced.

The scale factor can be calculated using:

scale factor = new length ÷ original length

For example, suppose a line that was originally:

4 cm

becomes:

12 cm

The scale factor is:

12 ÷ 4 = 3

Therefore:

scale factor = 3

Every length has been multiplied by 3.

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Enlargements

An enlargement makes a figure larger while keeping the same proportions.

If the scale factor is:

greater than 1

the figure is enlarged.

For example:

scale factor = 2

means every length is doubled.

A rectangle measuring:

4 cm × 3 cm

becomes:

8 cm × 6 cm

because:

4 × 2 = 8

and:

3 × 2 = 6

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Reductions

A reduction makes a figure smaller while keeping the same proportions.

If the scale factor is:

between 0 and 1

the figure is reduced.

For example:

scale factor = 0.5

means every length becomes half as long.

A rectangle measuring:

10 cm × 6 cm

becomes:

5 cm × 3 cm

because:

10 × 0.5 = 5

and:

6 × 0.5 = 3

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Scale Factor of 1

If:

scale factor = 1

the figure remains the same size.

For example:

8 cm × 1 = 8 cm

The figure has not been enlarged or reduced.

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Using Scale Factors

To calculate a new length:

new length = original length × scale factor

Example

A triangle has a side measuring:

7 cm

It is enlarged using a scale factor of:

3

Therefore:

new length = 7 × 3

new length = 21 cm

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Finding the Original Length

Sometimes the new measurement and scale factor are known.

Use:

original length = new length ÷ scale factor

Example

A figure has been enlarged by a scale factor of:

4

A side of the enlarged figure measures:

28 cm

Original length:

28 ÷ 4 = 7 cm

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Similar Figures

Two figures are similar when they have:

• The same shape.
• Equal corresponding angles.
• Corresponding side lengths in the same proportion.

Similar figures do not have to be the same size.

For example, two rectangles measuring:

4 cm × 6 cm

and:

8 cm × 12 cm

are similar.

Each side of the second rectangle is:

2 times

the corresponding side of the first.

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Similar Is Not the Same as Congruent

Congruent figures have:

• The same shape.
• The same size.

Similar figures have:

• The same shape.
• Not necessarily the same size.

All congruent figures are similar.

However, similar figures are not necessarily congruent.

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Corresponding Sides

Corresponding sides are sides that occupy matching positions in similar figures.

Suppose triangle ABC is similar to triangle DEF.

The corresponding vertices might be:

A ↔ D

B ↔ E

C ↔ F

Therefore, the corresponding sides are:

AB ↔ DE

BC ↔ EF

AC ↔ DF

It is important to match the correct sides before calculating a scale factor.

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Corresponding Angles

Corresponding angles in similar figures are equal.

For example, if:

∠A = 60°

then its corresponding angle:

∠D = 60°

Scaling changes lengths, but it does not change the angle measurements.

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Finding a Scale Factor

Suppose two similar triangles have corresponding sides:

small triangle = 5 cm

large triangle = 15 cm

Calculate:

scale factor = new ÷ original

scale factor = 15 ÷ 5

scale factor = 3

The larger triangle is therefore three times the size of the smaller triangle in each linear dimension.

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Finding Missing Lengths

Similarity allows us to determine unknown measurements.

Suppose two similar rectangles have:

Small rectangle:

width = 4 cm

length = 6 cm

Large rectangle:

width = 10 cm

length = x

First find the scale factor:

10 ÷ 4 = 2.5

Now apply the same scale factor to the length:

x = 6 × 2.5

x = 15 cm

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Using Proportions

Missing lengths can also be found using proportions.

Suppose two similar triangles have corresponding sides:

4 cm ↔ 10 cm

6 cm ↔ x

Write:

4/10 = 6/x

Cross multiply:

4x = 60

Therefore:

x = 15 cm

This works because corresponding sides of similar figures have equal ratios.

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Checking Whether Figures Are Similar

Consider two rectangles:

Rectangle A:

4 cm × 6 cm

Rectangle B:

8 cm × 12 cm

Compare corresponding sides:

8 ÷ 4 = 2

12 ÷ 6 = 2

The scale factor is the same.

Therefore, the rectangles are:

similar

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Example of Figures That Are Not Similar

Consider:

Rectangle A:

4 cm × 6 cm

Rectangle B:

8 cm × 10 cm

Compare corresponding sides:

8 ÷ 4 = 2

but:

10 ÷ 6 ≈ 1.67

The scale factors are different.

Therefore, the rectangles are:

not similar

Simply being the same type of shape does not always make two figures similar.

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Squares and Similarity

All squares are similar.

Why?

Every square has:

• Four 90° angles.
• Four equal sides.

A larger square is simply a scaled version of a smaller square.

For example:

A 3 cm × 3 cm square and a 12 cm × 12 cm square have scale factor:

12 ÷ 3 = 4

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Circles and Similarity

All circles are also similar.

A circle can be enlarged or reduced without changing its basic shape.

If the radius doubles, then:

• Radius doubles.
• Diameter doubles.
• Circumference doubles.

The circle remains similar to the original.

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Similar Triangles

Two triangles can have different sizes but still be similar.

For similar triangles:

• Corresponding angles are equal.
• Corresponding sides are proportional.

For example:

Triangle A has sides:

3 cm, 4 cm, 5 cm

Triangle B has sides:

6 cm, 8 cm, 10 cm

Each side has been multiplied by:

2

Therefore, the triangles are similar.

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Scale Drawings and Units

One important challenge with scale drawings is making sure the units match.

Suppose a map scale is:

1 cm : 5 km

A road measures:

7 cm

on the map.

Actual distance:

7 × 5 = 35 km

The answer should be written in:

kilometres

because that is the real-world unit given by the scale.

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Ratio Scales

Scale can also be written as a ratio.

For example:

1 : 100

This means:

1 unit on the drawing represents 100 of the same units in reality.

Therefore:

1 cm on drawing = 100 cm in reality

Since:

100 cm = 1 m

a 5 cm line on the drawing represents:

500 cm = 5 m

in reality.

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Understanding 1 : 50

A scale of:

1 : 50

means the real object is 50 times larger than the drawing.

If a wall measures:

8 cm

on the drawing:

actual length = 8 × 50

= 400 cm

Convert:

400 cm = 4 m

Therefore, the actual wall is:

4 m long

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Understanding 1 : 100

A scale of:

1 : 100

is particularly convenient.

It means:

1 cm on drawing = 100 cm in reality

Since:

100 cm = 1 m

we can also say:

1 cm on drawing = 1 m in reality

Therefore, a 7.5 cm wall on a 1:100 floor plan represents:

7.5 m

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Finding a Drawing Length

Suppose a building is:

30 m long

and the scale is:

1 cm : 5 m

Drawing length:

30 ÷ 5 = 6 cm

Therefore, the building should be drawn:

6 cm long

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Finding an Actual Length

Suppose a road measures:

9 cm

on a map with scale:

1 cm : 4 km

Actual length:

9 × 4 = 36 km

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Finding the Scale

Sometimes the drawing and actual measurements are known.

Suppose:

2 cm on a drawing represents 10 m

Divide both values by 2:

1 cm represents 5 m

Therefore, the scale can be written:

1 cm : 5 m

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Scale Drawings of Rooms

Architects and designers often use scale drawings of rooms and buildings.

Suppose a classroom measures:

8 m × 6 m

A drawing uses the scale:

1 cm : 1 m

The drawing dimensions will be:

8 cm × 6 cm

Furniture and other objects can then be placed proportionally on the plan.

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Maps

Maps are scale drawings of real geographic areas.

A map might use:

1 cm : 20 km

Suppose two cities are:

6.5 cm apart

on the map.

Actual distance:

6.5 × 20

= 130 km

Therefore, the map represents a distance of approximately:

130 km

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Blueprints

A blueprint or technical plan represents an object or structure using accurate scaled measurements.

Blueprints are used for:

• Buildings.
• Machines.
• Bridges.
• Furniture.
• Electrical systems.
• Engineering projects.

Engineers and builders can determine actual measurements from the drawing's scale.

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Models

Models also use scale.

For example, a model car might have a scale of:

1 : 20

This means the real car is:

20 times larger

in each linear dimension.

If the model car is:

22 cm long

then the real car is:

22 × 20 = 440 cm

or:

4.4 m

long.

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Scale Model Aircraft

Suppose a model airplane uses a scale of:

1 : 50

The real airplane has a wingspan of:

30 m

Convert to centimetres:

30 m = 3000 cm

Then divide by 50:

3000 ÷ 50 = 60 cm

The model should have a wingspan of:

60 cm

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Enlarging Small Objects

Scale drawings can also make very small objects easier to examine.

For example, diagrams of:

• Cells.
• Insects.
• Electronic components.
• Machine parts.

may be much larger than the actual objects.

If a drawing uses a scale factor of:

10

then every length shown is 10 times the corresponding actual length.

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Scale Factor and Perimeter

When all lengths are multiplied by a scale factor, the perimeter changes by the same factor.

Suppose a rectangle has perimeter:

20 cm

and is enlarged by scale factor:

3

New perimeter:

20 × 3 = 60 cm

This happens because every side length has been multiplied by 3.

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Scale Factor and Area

Area behaves differently.

Suppose a square has sides:

2 cm

Its area is:

2 × 2 = 4 cm²

Enlarge it using scale factor:

3

New side:

2 × 3 = 6 cm

New area:

6 × 6 = 36 cm²

Compare:

36 ÷ 4 = 9

The length scale factor was:

3

but the area scale factor was:

3² = 9

Therefore:

area scale factor = (length scale factor)²

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Why Area Changes Faster

Imagine a square enlarged by scale factor 2.

Original:

3 cm × 3 cm

Area:

9 cm²

Enlarged:

6 cm × 6 cm

Area:

36 cm²

The side lengths doubled, but:

36 ÷ 9 = 4

The area became four times larger.

This is because:

2² = 4

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Scale Factor and Volume

For three-dimensional similar objects, volume changes even faster.

If the length scale factor is:

k

then:

volume scale factor = k³

For example, if every dimension doubles:

volume scale factor = 2³ = 8

The new object has eight times the volume.

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Comparing Linear, Area, and Volume Scale Factors

If the linear scale factor is:

k

then:

For example, if:

k = 3

then:

• Lengths become 3 times larger.
• Perimeter becomes 3 times larger.
• Area becomes 9 times larger.
• Volume becomes 27 times larger.

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A Reliable Scale Drawing Strategy

When solving a scale drawing problem:

Step 1: Identify the scale.

Step 2: Identify whether you are finding the drawing measurement or actual measurement.

Step 3: Make sure the units are compatible.

Step 4: Multiply or divide using the scale.

Step 5: Include the correct units.

Step 6: Check whether the answer is reasonable.

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A Reliable Similar-Figures Strategy

When working with similar figures:

Step 1: Identify corresponding sides.

Step 2: Find the scale factor using a pair of known corresponding sides.

Step 3: Apply the same scale factor to the unknown side.

Step 4: Check that all corresponding lengths remain proportional.

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Worked Example: Map

A map uses:

1 cm : 8 km

Two towns are:

5.5 cm apart

on the map.

Actual distance:

5.5 × 8

= 44 km

Therefore:

The towns are 44 km apart.

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Worked Example: Blueprint

A blueprint uses a scale of:

1 : 50

A door measures:

4 cm

on the blueprint.

Actual measurement:

4 × 50 = 200 cm

Convert:

200 cm = 2 m

Therefore, the actual door is:

2 m

high or wide, depending on the measurement shown.

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Worked Example: Similar Triangles

Two triangles are similar.

The smaller triangle has corresponding sides:

4 cm and 7 cm

The larger triangle has:

12 cm and x

Find the scale factor:

12 ÷ 4 = 3

Therefore:

x = 7 × 3

x = 21 cm

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Worked Example: Reduction

A photograph measures:

24 cm × 18 cm

It is reduced using scale factor:

0.5

New length:

24 × 0.5 = 12 cm

New width:

18 × 0.5 = 9 cm

Therefore, the reduced photograph measures:

12 cm × 9 cm

The proportions have not changed.

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Common Mistakes

Using Different Scale Factors for Different Sides

For similar figures, the same scale factor must apply to every corresponding length.

If one side doubles while another triples, the figures are not similar.

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Matching the Wrong Sides

Always identify corresponding sides before calculating.

A long side should correspond to the matching long side, not automatically to whichever measurement appears first.

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Multiplying When You Should Divide

If you know the drawing measurement and need the larger real measurement, you will often multiply.

If you know the real measurement and need the smaller drawing measurement, you will often divide.

Think about whether your answer should become larger or smaller.

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Ignoring Units

Consider:

1 cm : 5 m

This does not mean:

1 : 5

because the units are different.

If writing a unitless ratio, convert to the same units first.

Since:

5 m = 500 cm

the ratio is:

1 : 500

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Assuming Similar Means Congruent

Similar figures have the same:

shape

but may have different:

sizes

Congruent figures have both the same size and shape.

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Changing Angles During Scaling

Enlargements and reductions change lengths.

They do not change corresponding angle measurements.

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Applying the Length Scale Factor Directly to Area

If lengths are multiplied by 3, area is not multiplied by 3.

It is multiplied by:

3² = 9

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Forgetting to Check Reasonableness

If a map uses:

1 cm : 10 km

and a 4 cm distance is calculated as 0.4 km, something has gone wrong.

We expect the real distance to be much larger than the map measurement.

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Check Your Understanding

1. Define a scale drawing.

2. What information does a scale provide?

3. Define scale factor.

4. What happens when the scale factor is greater than 1?

5. What happens when the scale factor is between 0 and 1?

6. What happens when the scale factor is exactly 1?

7. A 5 cm side is enlarged using scale factor 3. Find the new length.

8. A 12 cm side is reduced using scale factor 0.5. Find the new length.

9. A side changes from 4 cm to 20 cm. Find the scale factor.

10. A side becomes 24 cm after an enlargement with scale factor 3. Find its original length.

11. Define similar figures.

12. What happens to corresponding angles in similar figures?

13. What happens to corresponding side lengths in similar figures?

14. Explain the difference between similar and congruent figures.

15. Are all squares similar? Explain.

16. Are all rectangles similar? Explain.

17. Two similar figures have corresponding sides of 3 cm and 12 cm. Find the scale factor from the smaller figure to the larger figure.

18. A second side measures 5 cm on the smaller figure. Find its corresponding length on the larger figure from Question 17.

19. A map uses the scale 1 cm : 6 km. Two locations are 8 cm apart. Find their represented distance.

20. On the same map, two locations are 42 km apart. How far apart should they be drawn?

21. A blueprint uses a scale of 1:100. A wall measures 7 cm on the plan. Find its actual length in metres.

22. A model car uses a scale of 1:25. The model is 18 cm long. Find the length of the real car.

23. A real building is 40 m long. A plan uses 1 cm : 5 m. How long should the building appear on the plan?

24. Two rectangles measure 4 cm × 6 cm and 10 cm × 15 cm. Are they similar? Explain.

25. Two rectangles measure 4 cm × 6 cm and 8 cm × 10 cm. Are they similar? Explain.

26. A figure is enlarged using scale factor 4. By what factor does its perimeter increase?

27. A figure is enlarged using scale factor 4. By what factor does its area increase?

28. A three-dimensional model is enlarged using scale factor 2. By what factor does its volume increase?

29. Explain why scale drawings are useful for architects and engineers.

30. Give three real-world situations in which scale drawings or similar figures are useful.

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Key Terms

• Scale drawing – representation of an object in which all measurements are changed proportionally.
• Scale – relationship between a measurement on a drawing and the corresponding actual measurement.
• Scale factor – number by which corresponding lengths are multiplied.
• Enlargement – proportional increase in the size of a figure.
• Reduction – proportional decrease in the size of a figure.
• Similar figures – figures with the same shape, equal corresponding angles, and proportional corresponding sides.
• Congruent figures – figures with the same size and shape.
• Corresponding sides – sides that occupy matching positions in similar figures.
• Corresponding angles – angles that occupy matching positions in similar figures.
• Proportion – equality between two ratios.
• Blueprint – technical scale drawing used in design or construction.
• Model – representation of an object, often produced using a particular scale.

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Key Takeaways

• A scale drawing represents a real object at a proportional size.
• The scale connects measurements on the drawing with actual measurements.
• A scale factor describes how much lengths have been multiplied.
• A scale factor greater than 1 produces an enlargement.
• A scale factor between 0 and 1 produces a reduction.
• Similar figures have the same shape but may have different sizes.
• Corresponding angles of similar figures are equal.
• Corresponding sides of similar figures are proportional.
• The same scale factor applies to all corresponding lengths.
• Scale drawings can be used to calculate unknown real-world measurements.
• Units must be considered carefully when using scales.
• Maps, blueprints, floor plans, diagrams, and models all use scale.
• When the length scale factor is k, the perimeter scale factor is also k.
• The area scale factor is k².
• For similar three-dimensional objects, the volume scale factor is k³.

The central relationship to remember is:

new length = original length × scale factor

and for similar figures:

corresponding lengths must all change by the same scale factor.