Edexcel Mathematics (4XMAH) -Unit 2 - 4.11 Similarity- Study Notes- New Syllabus
Edexcel Mathematics (4XMAH) -Unit 2 – 4.11 Similarity- Study Notes- New syllabus
Edexcel Mathematics (4XMAH) -Unit 2 – 4.11 Similarity- Study Notes -Edexcel iGCSE Mathematics – per latest Syllabus.
Key Concepts:
A areas of similar figures are in the ratio of the squares of corresponding sides
B volumes of similar figures are in the ratio of the cubes of corresponding sides
C use areas and volumes of similar figures in solving problems
Edexcel iGCSE Mathematics -Concise Summary Notes- All Topics
Areas of Similar Figures
Two shapes are similar if they have the same shape but different size.
Their corresponding angles are equal and their corresponding sides are in the same ratio.
If the scale factor between two similar shapes is \( k \), then every length in the larger shape is \( k \) times the smaller shape.
Key Rule for Area
The ratio of the areas of two similar figures equals the square of the ratio of their corresponding sides.
If side ratio \( =a:b \)
Area ratio \( =a^2:b^2 \)
This happens because area depends on two dimensions (length × width).
Example of Scale Factor
If a triangle is enlarged with scale factor 3:
- All sides triple
- Area becomes \( 3^2=9 \) times larger
Important Idea
Do not multiply areas directly by the side ratio. You must square the ratio first.
Example 1:
Two similar squares have side lengths 2 cm and 6 cm. Find the ratio of their areas.
▶️ Answer/Explanation
Side ratio:
\( 2:6=1:3 \)
Area ratio:
\( 1^2:3^2=1:9 \)
Conclusion: \( 1:9 \).
Example 2:
Two similar triangles have corresponding sides in the ratio \( 3:5 \). Find the ratio of their areas.
▶️ Answer/Explanation
Area ratio \( =3^2:5^2=9:25 \)
Conclusion: \( 9:25 \).
Example 3:
A rectangle is enlarged by scale factor 4. By what factor does its area increase?
▶️ Answer/Explanation
Area factor \( =4^2=16 \)
Conclusion: Area becomes 16 times larger.
Volumes of Similar Figures
When 3D shapes are similar, they have the same shape but different size.
Their corresponding lengths are in a constant ratio called the scale factor.
If the scale factor between two similar solids is \( k \):
- All lengths multiply by \( k \)
- Areas multiply by \( k^2 \)
- Volumes multiply by \( k^3 \)
Key Rule for Volume
The ratio of the volumes of similar solids equals the cube of the ratio of corresponding sides.
If side ratio \( =a:b \)
Volume ratio \( =a^3:b^3 \)
This happens because volume depends on three dimensions: length, width and height.
Important Reminder
Students often square the ratio by mistake. For volume you must cube the ratio.
Example 1:
Two similar cubes have side lengths 2 cm and 6 cm. Find the ratio of their volumes.
▶️ Answer/Explanation
Side ratio:
\( 2:6=1:3 \)
Volume ratio:
\( 1^3:3^3=1:27 \)
Conclusion: \( 1:27 \).
Example 2:
Two similar cones have radii in the ratio \( 3:5 \). Find the ratio of their volumes.
▶️ Answer/Explanation
Volume ratio \( =3^3:5^3=27:125 \)
Conclusion: \( 27:125 \).
Example 3:
A solid is enlarged with scale factor 4. By what factor does the volume increase?
▶️ Answer/Explanation
Volume factor \( =4^3=64 \)
Conclusion: Volume becomes 64 times larger.
Using Areas and Volumes of Similar Figures in Problems
In many questions you are not given the scale factor directly. Instead, you must use area or volume information to find missing lengths.
Key Relationships
- Side ratio \( =k \)
- Area ratio \( =k^2 \)
- Volume ratio \( =k^3 \)
So:
- To go from area to length → take square root
- To go from volume to length → take cube root
Important Skill
Always convert area or volume ratios back to a side ratio before finding a missing length.
Example 1:
Two similar triangles have areas in the ratio \( 16:25 \). Find the ratio of their corresponding sides.
▶️ Answer/Explanation
Side ratio is square root of area ratio:
\( \sqrt{16:25}=4:5 \)
Conclusion: Side ratio \( 4:5 \).
Example 2:
Two similar cones have volumes in the ratio \( 8:27 \). Find the ratio of their heights.
▶️ Answer/Explanation
Height ratio equals side ratio.
Take cube root:
\( \sqrt[3]{8:27}=2:3 \)
Conclusion: Height ratio \( 2:3 \).
Example 3:
A model pyramid is similar to a real pyramid. The model height is 5 cm and the real pyramid volume is 1000 times larger. Find the height of the real pyramid.
▶️ Answer/Explanation
Volume ratio \( =1000:1 \)
Side ratio is cube root:
\( \sqrt[3]{1000}=10 \)
Real height:
\( 5\times10=50\text{ cm} \)
Conclusion: \( 50\text{ cm} \).