MathematicsChapter 9: Geometry and Polygons
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.
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 are proportional. A scale factor 2 means areas x4 and volumes x8, not x2.
Overview
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.
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.
Examples Two spheres radii 2 cm and 6 cm: k=3, volumes 1:27 · Paint needed ~ surface · Water to fill ~ volume
Explanation
Regular polygons and angle sums were an earlier page. Similarity is the extra tool when shapes are the same but sized differently.
In simple words
Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
Key points
- lengths k
- areas k^2
- volumes k^3
- perimeter with k
- area with k^2
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Similarity: Area and Volume Ratios
Sticky Notes · Quick Revision
Core Idea
- If two figures are similar with length ratio k (bigger/smaller), area ratio is k^2 and volume ratio is k^3.
Key Points
- lengths k
- areas k^2
- volumes k^3
Simple Check
- Explain similarity: area and volume ratios in your own words and connect it to one chapter example.
How to Answer
- Define the idea, show the main rule or evidence, apply it, and check the final statement.
Common Mistake
- Congruent means k=1: same size and shape. Similar allows k not 1. All congruent figures are similar; the reverse is false.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- lengths k
- areas k^2
- volumes k^3
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