Scale Factors and Maps

Mathematically Similar Shapes

When you zoom in on a photo on your phone, the image gets larger but the picture doesn't distort. This is because the original picture and the zoomed-in picture are mathematically similar.

Two shapes are mathematically similar if they are the exact same shape but different sizes. In similar shapes, all corresponding angles remain identical—they do not change size—while all corresponding sides are in the same ratio. The multiplier used to change the size of the shape's lengths is called the linear scale factor ( $k$ ).

An object is the original shape before a transformation, and the image is the shape after the transformation has been applied. The transformation is measured from a fixed point called the centre of enlargement.

If the scale factor is greater than $1$ , the shape gets larger, which is called an enlargement. If the scale factor is between $0$ and $1$ , the shape gets smaller, which is known as a reduction.

Calculating the Linear Scale Factor

To find the linear scale factor, you compare the lengths of matching sides on both shapes.

\text{Scale Factor} = \frac{\text{New Length}}{\text{Original Length}}

To find a missing length on the new, larger shape, multiply the original length by the scale factor. To find a missing length on the smaller shape, divide the larger length by the scale factor.

Worked Example: Finding a Missing Side Length

Triangle A (the object) has a base of $6$ cm and a height of $4$ cm. Triangle B (the image) is mathematically similar and has a base of $15$ cm. Calculate the height ( $x$ ) of Triangle B.

Step 1: Calculate the linear scale factor using the corresponding base lengths.

$k = \frac{15}{6} = 2.5$

Step 2: Apply the scale factor to the known height of Triangle A.

$x = 4 \times 2.5$

Step 3: State the final answer with units.

$x = 10$ cm

Scale Diagrams and Map Conversions

Why does a map of the UK fit onto a single piece of paper while representing hundreds of miles in reality? Maps use a ratio scale (like $1:n$ ) to scale down real-world distances perfectly.

A ratio scale means that $1$ unit of map distance represents $n$ units of actual distance in real life. This ratio is often written as a representative fraction (such as $\frac{1}{50,000}$ ). The ratio has no units until you apply the exact same units to both sides (e.g., $1$ cm on the map equals $50,000$ cm in reality).

When calculating map distances, you must be confident converting between metric units.

$1$ cm = $10$ mm
$1$ m = $100$ cm
$1$ km = $1,000$ m
$1$ km = $100,000$ cm

To convert from map distance to real-life distance, multiply by the scale factor. To convert from real-life distance back to the map, divide by the scale factor.

Worked Example: Map to Real Distance

A map scale is $1:50,000$ . The distance between two towns on the map is $8.5$ cm. Calculate the actual distance between the towns in kilometres.

Step 1: Multiply the map distance by the ratio scale factor to find the real distance in cm.

$\text{Actual distance} = 8.5 \times 50,000 = 425,000$ cm

Step 2: Convert centimetres to metres by dividing by $100$ .

$425,000 \div 100 = 4,250$ m

Step 3: Convert metres to kilometres by dividing by $1,000$ .

$4,250 \div 1,000 = 4.25$ km

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Exam Tips

Common Mistake
Students often fall into the 'addition trap' by adding the difference between sides (e.g., adding 9 cm because 15 - 6 = 9) instead of multiplying by the scale factor.
2
In Edexcel exams, the phrase 'mathematically similar' is your cue that you must use scale factor methods.
3
Always write down the initial fraction for your scale factor (e.g., 15/6), as examiners frequently award a method mark (M1) for this even if your final calculation is wrong.
4
Do not round a linear scale factor too early in your working (e.g., using 1.3 instead of 1.333...), as this will lead to an inaccurate final length.
Common Mistake
When converting actual distances from centimetres straight to kilometres, students often wrongly divide by 1,000; remember that 1 km = 100,000 cm, so you must divide by 100,000.

Key Terms(12)

Mathematically similar: Two shapes are mathematically similar if they are the exact same shape but different sizes, with equal corresponding angles and side lengths in the same ratio.
Corresponding sides: Sides that are in the exact same relative position on two mathematically similar shapes.
Linear scale factor: The multiplier used to change the size of a shape's lengths during an enlargement or reduction.
Object: The original shape before a transformation (such as an enlargement or reduction) is applied.
Image: The new shape produced after a transformation has been applied to an object.
Centre of enlargement: A fixed coordinate point from which an enlargement or reduction is measured.
Enlargement: A transformation where the scale factor is greater than 1, resulting in an image that is larger than the object.
Reduction: A transformation where the scale factor is a fraction between 0 and 1, resulting in an image that is smaller than the object.
Ratio scale: A dimensionless scale (e.g., 1:50,000) indicating that 1 unit on a map represents a specific number of units in reality.
Representative fraction: A map scale expressed as a fraction or ratio where the first number is always 1.
Map distance: The length between two points measured on a map, usually recorded in centimetres or millimetres.
Actual distance: The real-world distance between two points, usually expressed in metres or kilometres.

Previous Subtopic: Units and Compound UnitsUnits and Compound Units Next Subtopic: Fractional QuantitiesFractional Quantities

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Key Terms(12)

Mathematically similar: Two shapes are mathematically similar if they are the exact same shape but different sizes, with equal corresponding angles and side lengths in the same ratio.
Corresponding sides: Sides that are in the exact same relative position on two mathematically similar shapes.
Linear scale factor: The multiplier used to change the size of a shape's lengths during an enlargement or reduction.
Object: The original shape before a transformation (such as an enlargement or reduction) is applied.
Image: The new shape produced after a transformation has been applied to an object.
Centre of enlargement: A fixed coordinate point from which an enlargement or reduction is measured.
Enlargement: A transformation where the scale factor is greater than 1, resulting in an image that is larger than the object.
Reduction: A transformation where the scale factor is a fraction between 0 and 1, resulting in an image that is smaller than the object.
Ratio scale: A dimensionless scale (e.g., 1:50,000) indicating that 1 unit on a map represents a specific number of units in reality.
Representative fraction: A map scale expressed as a fraction or ratio where the first number is always 1.
Map distance: The length between two points measured on a map, usually recorded in centimetres or millimetres.
Actual distance: The real-world distance between two points, usually expressed in metres or kilometres.

Scale Factors and Maps

Mathematically Similar Shapes

When you zoom in on a photo on your phone, the image gets larger but the picture doesn't distort. This is because the original picture and the zoomed-in picture are mathematically similar.

Calculating the Linear Scale Factor

To find the linear scale factor, you compare the lengths of matching sides on both shapes.

\text{Scale Factor} = \frac{\text{New Length}}{\text{Original Length}}

To find a missing length on the new, larger shape, multiply the original length by the scale factor. To find a missing length on the smaller shape, divide the larger length by the scale factor.

Worked Example: Finding a Missing Side Length

Triangle A (the object) has a base of $6$ cm and a height of $4$ cm. Triangle B (the image) is mathematically similar and has a base of $15$ cm. Calculate the height ( $x$ ) of Triangle B.

Step 1: Calculate the linear scale factor using the corresponding base lengths.

$k = \frac{15}{6} = 2.5$

Step 2: Apply the scale factor to the known height of Triangle A.

$x = 4 \times 2.5$

Step 3: State the final answer with units.

$x = 10$ cm

Scale Diagrams and Map Conversions

Why does a map of the UK fit onto a single piece of paper while representing hundreds of miles in reality? Maps use a ratio scale (like $1:n$ ) to scale down real-world distances perfectly.

When calculating map distances, you must be confident converting between metric units.

$1$ cm = $10$ mm
$1$ m = $100$ cm
$1$ km = $1,000$ m
$1$ km = $100,000$ cm

To convert from map distance to real-life distance, multiply by the scale factor. To convert from real-life distance back to the map, divide by the scale factor.

Worked Example: Map to Real Distance

A map scale is $1:50,000$ . The distance between two towns on the map is $8.5$ cm. Calculate the actual distance between the towns in kilometres.

Step 1: Multiply the map distance by the ratio scale factor to find the real distance in cm.

$\text{Actual distance} = 8.5 \times 50,000 = 425,000$ cm

Step 2: Convert centimetres to metres by dividing by $100$ .

$425,000 \div 100 = 4,250$ m

Step 3: Convert metres to kilometres by dividing by $1,000$ .

$4,250 \div 1,000 = 4.25$ km

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Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often fall into the 'addition trap' by adding the difference between sides (e.g., adding 9 cm because 15 - 6 = 9) instead of multiplying by the scale factor.
2
In Edexcel exams, the phrase 'mathematically similar' is your cue that you must use scale factor methods.
3
Always write down the initial fraction for your scale factor (e.g., 15/6), as examiners frequently award a method mark (M1) for this even if your final calculation is wrong.
4
Do not round a linear scale factor too early in your working (e.g., using 1.3 instead of 1.333...), as this will lead to an inaccurate final length.
Common Mistake
When converting actual distances from centimetres straight to kilometres, students often wrongly divide by 1,000; remember that 1 km = 100,000 cm, so you must divide by 100,000.

Key Terms(12)

Mathematically similar: Two shapes are mathematically similar if they are the exact same shape but different sizes, with equal corresponding angles and side lengths in the same ratio.
Corresponding sides: Sides that are in the exact same relative position on two mathematically similar shapes.
Linear scale factor: The multiplier used to change the size of a shape's lengths during an enlargement or reduction.
Object: The original shape before a transformation (such as an enlargement or reduction) is applied.
Image: The new shape produced after a transformation has been applied to an object.
Centre of enlargement: A fixed coordinate point from which an enlargement or reduction is measured.
Enlargement: A transformation where the scale factor is greater than 1, resulting in an image that is larger than the object.
Reduction: A transformation where the scale factor is a fraction between 0 and 1, resulting in an image that is smaller than the object.
Ratio scale: A dimensionless scale (e.g., 1:50,000) indicating that 1 unit on a map represents a specific number of units in reality.
Representative fraction: A map scale expressed as a fraction or ratio where the first number is always 1.
Map distance: The length between two points measured on a map, usually recorded in centimetres or millimetres.
Actual distance: The real-world distance between two points, usually expressed in metres or kilometres.

Previous Subtopic: Units and Compound UnitsUnits and Compound Units Next Subtopic: Fractional QuantitiesFractional Quantities

Back to Scale Factors and Maps

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(12)

Mathematically similar: Two shapes are mathematically similar if they are the exact same shape but different sizes, with equal corresponding angles and side lengths in the same ratio.
Corresponding sides: Sides that are in the exact same relative position on two mathematically similar shapes.
Linear scale factor: The multiplier used to change the size of a shape's lengths during an enlargement or reduction.
Object: The original shape before a transformation (such as an enlargement or reduction) is applied.
Image: The new shape produced after a transformation has been applied to an object.
Centre of enlargement: A fixed coordinate point from which an enlargement or reduction is measured.
Enlargement: A transformation where the scale factor is greater than 1, resulting in an image that is larger than the object.
Reduction: A transformation where the scale factor is a fraction between 0 and 1, resulting in an image that is smaller than the object.
Ratio scale: A dimensionless scale (e.g., 1:50,000) indicating that 1 unit on a map represents a specific number of units in reality.
Representative fraction: A map scale expressed as a fraction or ratio where the first number is always 1.
Map distance: The length between two points measured on a map, usually recorded in centimetres or millimetres.
Actual distance: The real-world distance between two points, usually expressed in metres or kilometres.