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Similarity: Area and Volume Ratios
If two figures are similar with length ratio k (bigger/smaller), area ratio is k^2 and volume ratio is k^3. Corresponding angles are equal. Corresponding sides are proportional. A scale factor 2 means areas x4 and volumes x8, not x2.
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Similarity: Area and Volume Ratios
What is it?
If two figures are similar with length ratio k (bigger/smaller), area ratio is k^2 and volume ratio is k^3. Corresponding angles are equal. Corresponding sides are proportional. A scale factor 2 means areas x4 and volumes x8, not x2.
In simple words
Similarity: Area and Volume Ratios is a core idea in this chapter. Learn the meaning first, then the main parts, then one clear example.
Examples
- Map scale 1:100 000
- Two similar triangles 3-4-5 and 6-8-10
- Do not square an angle
LENGTH AND AREA
Triangles: AA similarity is enough in Class 9. If sides 3:6, k=2, areas 1:4. Height, median, perimeter all scale with k. Area of similar polygons: (side ratio)^2.
VOLUME AND SURFACE
Similar solids (cubes, spheres, named pyramids): surface like area (k^2), volume k^3. A cube of side double has volume eight times, surface four times.
Recommended for Class 9
Full school-level explanation
If two figures are similar with length ratio k (bigger/smaller), area ratio is k^2 and volume ratio is k^3. Corresponding angles are equal. Corresponding sides are proportional. A scale factor 2 means areas x4 and volumes x8, not x2.
Linear first, then square or cube:
LENGTH AND AREA
Triangles: AA similarity is enough in Class 9. If sides 3:6, k=2, areas 1:4. Height, median, perimeter all scale with k. Area of similar polygons: (side ratio)^2.
Remember
- perimeter with k
- area with k^2
Examples
- Map scale 1:100 000
- Two similar triangles 3-4-5 and 6-8-10
- Do not square an angle
VOLUME AND SURFACE
Similar solids (cubes, spheres, named pyramids): surface like area (k^2), volume k^3. A cube of side double has volume eight times, surface four times.
Remember
- surface k^2
- volume k^3
Examples
- Two spheres radii 2 cm and 6 cm: k=3, volumes 1:27
- Paint needed ~ surface
- Water to fill ~ volume
Remember
- lengths k
- areas k^2
- volumes k^3
- same shape, different size
- congruent => similar
- perimeter with k
- area with k^2
- surface k^2
How the ideas connect
Regular polygons and angle sums were an earlier page. Similarity is the extra tool when shapes are the same but sized differently.
Deeper understanding
Go beyond memorising Similarity: Area and Volume Ratios. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Regular polygons and angle sums were an earlier page. Similarity is the extra tool when shapes are the same but sized differently.
Exam focus
Quick revision
- lengths k
- areas k^2
- volumes k^3
- same shape, different size
- LENGTH AND AREA: Triangles: AA similarity is enough in Class 9. If sides 3:6, k=2, areas 1:4. Height, median, perimet
- VOLUME AND SURFACE: Similar solids (cubes, spheres, named pyramids): surface like area (k^2), volume k^3. A cube of side
Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
For long answers on Similarity: Area and Volume Ratios: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- lengths k
- areas k^2
- volumes k^3
- same shape, different size
Important formulas
- lengths k
- areas k^2
- volumes k^3
Common exam traps
- Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
- Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
- Mistake
Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
- Mistake
Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
lengths k
areas k^2
volumes k^3
same shape, different size
If two figures are similar with length ratio k (bigger/smaller), area ratio is k^2 and volume ratio is k^3. Corresponding angles are equal. Corresponding sides
congruent => similar
- Definition
- If two figures are similar with length ratio k (bigger/smaller), area ratio is k^2 and volume ratio is k^3. Corresponding angles are equal. Corresponding sides
- Formula
- congruent => similar
- Example
- Map scale 1:100 000
- Critical warning
- Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is fals
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Exam Practice
Define
Q1. Define Similarity: Area and Volume Ratios.
Explain
Q2. Explain: LENGTH AND AREA.
Example
Q3. Give one clear example of Similarity: Area and Volume Ratios and say why it fits.
Mistake
Q4. State one common mistake students make about Similarity: Area and Volume Ratios.
Exam note
Q5. Write a short exam note on Similarity: Area and Volume Ratios: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.