Similarity Calculations

Visualising Similarity

Imagine taking a photograph on your phone and zooming in—everything gets larger, but the shape stays exactly the same. This is mathematical similarity in action, where every length is stretched by the exact same multiplier. However, as you will see, the area and volume of the shape grow much faster than the lengths do!

Congruence and Similarity

Mathematically Similar shapes have the exact same shape but are different sizes, meaning all corresponding angles are equal.
To prove two triangles are similar, you only need to show all three angles are equal (AAA). However, similar shapes do NOT necessarily have the same size, so they are not automatically congruent.
Congruent shapes are identical in both shape and size.
For congruent shapes, the scale factor is exactly $k = 1$ , which means all corresponding lengths, areas, and volumes are completely identical.

The Scale Factor Rules

The Linear Scale Factor ( $k$ ) is the constant multiplier used to scale lengths between similar shapes.
It is calculated using:

k = \frac{\text{New Length}}{\text{Original Length}}

The Area Scale Factor is $k^2$ . This applies to any 2D area, cross-sectional area, or 3D total surface area.
The Volume Scale Factor is $k^3$ . This applies to volume, mass (if density is uniform), and Capacity.

Worked Example: Finding Area from Lengths

A small monster toy has a height of $20$ cm and a surface area of $300$ cm $^2$ . A mathematically similar large monster toy has a height of $120$ cm. Calculate the surface area of the large monster toy.

Step 1: Calculate the linear scale factor ( $k$ ).

$k = 120 \div 20 = 6$

Step 2: Calculate the Area Scale Factor ( $k^2$ ).

$k^2 = 6^2 = 36$

Step 3: Multiply the small area by the Area Scale Factor.

$\text{Large Area} = 300 \times 36 = 10{,}800$ cm $^2$

Worked Example: Finding Length from Volumes

Two mathematically similar cones have volumes of $20$ cm $^3$ and $540$ cm $^3$ . The height of the larger cone is $15$ cm. Calculate the height of the smaller cone.

Step 1: Calculate the Volume Scale Factor ( $k^3$ ).

$k^3 = 540 \div 20 = 27$

Step 2: Find the linear scale factor ( $k$ ) by cube rooting.

$k = \sqrt[3]{27} = 3$

Step 3: Divide the larger height by $k$ to find the smaller height.

$\text{Small Height} = 15 \div 3 = 5$ cm

Transitioning Dimensions (Higher Tier)

You cannot convert directly between area and volume scale factors.
You must always find the linear scale factor ( $k$ ) first by square rooting the Area Scale Factor or cube rooting the Volume Scale Factor.
For example, if the volumes of two similar solids are in the ratio $27:64$ , their lengths are in the ratio $\sqrt[3]{27}:\sqrt[3]{64}$ which simplifies to $3:4$ .
Their surface areas would then be in the ratio $3^2:4^2$ , giving $9:16$ .

Negative Scale Factors (Higher Tier)

A negative scale factor ( $k < 0$ ) means the shape is inverted and rotated $180^\circ$ through a center of enlargement.
Physical measurements like length, area, and volume cannot be negative.
To find a missing length when $k$ is negative, multiply by the absolute value $|k|$ .
Because any number squared is positive, a negative linear scale factor still results in a positive Area Scale Factor ( $k^2$ ).

Sign up to continue reading

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

Exam Tips

Common Mistake
Students frequently multiply volume by the linear scale factor ( $k$ ) instead of the volume scale factor ( $k^3$ ). Always remember to cube the scale factor for 3D measurements!
2
In AQA exams, the phrase 'mathematically similar' is your cue to write down $k$ , $k^2$ , and $k^3$ immediately to secure method marks.
3
You cannot jump directly from an area ratio to a volume ratio; you must always find the linear scale factor ( $k$ ) first by square rooting or cube rooting.
4
When calculating the scale factor, never round decimals mid-calculation (e.g., $1.333...$ ); use fractions or the 'Ans' button on your calculator to keep the exact value.

Key Terms(6)

Mathematically Similar: Shapes that have the same shape but are different sizes, where all corresponding angles are equal.
Congruent: Identical in shape and size, meaning all corresponding lengths and angles are exactly equal.
Linear Scale Factor (k): The constant multiplier used to change the size of a shape while maintaining its proportions.
Area Scale Factor: The ratio of two corresponding areas in similar figures, calculated by squaring the linear scale factor ( $k^2$ ).
Volume Scale Factor: The ratio of the volumes or capacities of two similar 3D solids, calculated by cubing the linear scale factor ( $k^3$ ).
Capacity: The amount a 3D object can hold, which is treated identically to volume in mathematical similarity problems.

Previous Subtopic: Sectors and ArcsSectors and Arcs Next Subtopic: Pythagoras and TrigonometryPythagoras and Trigonometry

Back to Similarity Calculations

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

Key Terms(6)

Mathematically Similar: Shapes that have the same shape but are different sizes, where all corresponding angles are equal.
Congruent: Identical in shape and size, meaning all corresponding lengths and angles are exactly equal.
Linear Scale Factor (k): The constant multiplier used to change the size of a shape while maintaining its proportions.
Area Scale Factor: The ratio of two corresponding areas in similar figures, calculated by squaring the linear scale factor ( $k^2$ ).
Volume Scale Factor: The ratio of the volumes or capacities of two similar 3D solids, calculated by cubing the linear scale factor ( $k^3$ ).
Capacity: The amount a 3D object can hold, which is treated identically to volume in mathematical similarity problems.

Similarity Calculations

Visualising Similarity

Congruence and Similarity

Mathematically Similar shapes have the exact same shape but are different sizes, meaning all corresponding angles are equal.
To prove two triangles are similar, you only need to show all three angles are equal (AAA). However, similar shapes do NOT necessarily have the same size, so they are not automatically congruent.
Congruent shapes are identical in both shape and size.
For congruent shapes, the scale factor is exactly $k = 1$ , which means all corresponding lengths, areas, and volumes are completely identical.

The Scale Factor Rules

The Linear Scale Factor ( $k$ ) is the constant multiplier used to scale lengths between similar shapes.
It is calculated using:

k = \frac{\text{New Length}}{\text{Original Length}}

The Area Scale Factor is $k^2$ . This applies to any 2D area, cross-sectional area, or 3D total surface area.
The Volume Scale Factor is $k^3$ . This applies to volume, mass (if density is uniform), and Capacity.

Worked Example: Finding Area from Lengths

Step 1: Calculate the linear scale factor ( $k$ ).

$k = 120 \div 20 = 6$

Step 2: Calculate the Area Scale Factor ( $k^2$ ).

$k^2 = 6^2 = 36$

Step 3: Multiply the small area by the Area Scale Factor.

$\text{Large Area} = 300 \times 36 = 10{,}800$ cm $^2$

Worked Example: Finding Length from Volumes

Two mathematically similar cones have volumes of $20$ cm $^3$ and $540$ cm $^3$ . The height of the larger cone is $15$ cm. Calculate the height of the smaller cone.

Step 1: Calculate the Volume Scale Factor ( $k^3$ ).

$k^3 = 540 \div 20 = 27$

Step 2: Find the linear scale factor ( $k$ ) by cube rooting.

$k = \sqrt[3]{27} = 3$

Step 3: Divide the larger height by $k$ to find the smaller height.

$\text{Small Height} = 15 \div 3 = 5$ cm

Transitioning Dimensions (Higher Tier)

You cannot convert directly between area and volume scale factors.
You must always find the linear scale factor ( $k$ ) first by square rooting the Area Scale Factor or cube rooting the Volume Scale Factor.
For example, if the volumes of two similar solids are in the ratio $27:64$ , their lengths are in the ratio $\sqrt[3]{27}:\sqrt[3]{64}$ which simplifies to $3:4$ .
Their surface areas would then be in the ratio $3^2:4^2$ , giving $9:16$ .

Negative Scale Factors (Higher Tier)

A negative scale factor ( $k < 0$ ) means the shape is inverted and rotated $180^\circ$ through a center of enlargement.
Physical measurements like length, area, and volume cannot be negative.
To find a missing length when $k$ is negative, multiply by the absolute value $|k|$ .
Because any number squared is positive, a negative linear scale factor still results in a positive Area Scale Factor ( $k^2$ ).

Sign up to continue reading

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

Exam Tips

Common Mistake
Students frequently multiply volume by the linear scale factor ( $k$ ) instead of the volume scale factor ( $k^3$ ). Always remember to cube the scale factor for 3D measurements!
2
In AQA exams, the phrase 'mathematically similar' is your cue to write down $k$ , $k^2$ , and $k^3$ immediately to secure method marks.
3
You cannot jump directly from an area ratio to a volume ratio; you must always find the linear scale factor ( $k$ ) first by square rooting or cube rooting.
4
When calculating the scale factor, never round decimals mid-calculation (e.g., $1.333...$ ); use fractions or the 'Ans' button on your calculator to keep the exact value.

Key Terms(6)

Mathematically Similar: Shapes that have the same shape but are different sizes, where all corresponding angles are equal.
Congruent: Identical in shape and size, meaning all corresponding lengths and angles are exactly equal.
Linear Scale Factor (k): The constant multiplier used to change the size of a shape while maintaining its proportions.
Area Scale Factor: The ratio of two corresponding areas in similar figures, calculated by squaring the linear scale factor ( $k^2$ ).
Volume Scale Factor: The ratio of the volumes or capacities of two similar 3D solids, calculated by cubing the linear scale factor ( $k^3$ ).
Capacity: The amount a 3D object can hold, which is treated identically to volume in mathematical similarity problems.

Previous Subtopic: Sectors and ArcsSectors and Arcs Next Subtopic: Pythagoras and TrigonometryPythagoras and Trigonometry

Back to Similarity Calculations

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

Key Terms(6)

Mathematically Similar: Shapes that have the same shape but are different sizes, where all corresponding angles are equal.
Congruent: Identical in shape and size, meaning all corresponding lengths and angles are exactly equal.
Linear Scale Factor (k): The constant multiplier used to change the size of a shape while maintaining its proportions.
Area Scale Factor: The ratio of two corresponding areas in similar figures, calculated by squaring the linear scale factor ( $k^2$ ).
Volume Scale Factor: The ratio of the volumes or capacities of two similar 3D solids, calculated by cubing the linear scale factor ( $k^3$ ).
Capacity: The amount a 3D object can hold, which is treated identically to volume in mathematical similarity problems.