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Area ratio of similar triangles

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides.

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Student-friendly explanation

Area grows faster than side length. When every side is scaled by the same factor, the area changes by the square of that factor, not by the same factor itself.

How to write this in exams

  1. 1

    Start with the exact idea

    For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides.

  2. 2

    Then show how to use it

    1. Write the side ratio. 2. Square both terms. 3. Simplify the ratio if needed. 4. Use it for area comparison.

  3. 3

    Add one concrete example

    If the side ratio is 2:3, the area ratio is 4:9.

  4. 4

    Avoid this incomplete answer

    Using 3:5 as the area ratio directly and ignoring the square relationship.

Definition

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides.

Example

If the side ratio is 2:3, the area ratio is 4:9.

Rule to remember

Area ratio of similar triangles = (corresponding side ratio)^2.

Memory hook

Sides scale once, areas scale twice.

Examples and method

Worked example

If similar triangles have sides in the ratio 3:5, their areas are in the ratio 9:25. If one area is 36 cm², the other becomes 100 cm² when the smaller triangle matches the 3:5 scale.

Method to apply

1. Write the side ratio. 2. Square both terms. 3. Simplify the ratio if needed. 4. Use it for area comparison.

Diagram support

Mark matching sides clearly before squaring the ratio.

How CBSE asks it

Questions may give side ratios and ask for area ratio, or give area ratio and ask for side ratio.

Avoid common mistakes

Common confusion

Students often keep the same ratio for area instead of squaring it.

Common wrong answer

Using 3:5 as the area ratio directly and ignoring the square relationship.

Exam tip

Find the side ratio first, then square both terms carefully.

Quick check

How is the area ratio of two similar triangles related to their side ratio?

The area ratio is the square of the corresponding side ratio, not the same ratio. So if the sides are in the ratio 3:5, the areas are in the ratio 9:25.

Answer writing and exam use

1-mark answer

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides.

2-mark answer

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides. Area ratio of similar triangles = (corresponding side ratio)^2. If the side ratio is 2:3, the area ratio is 4:9.

3-mark answer

Area grows faster than side length. When every side is scaled by the same factor, the area changes by the square of that factor, not by the same factor itself. Area ratio of similar triangles = (corresponding side ratio)^2. If similar triangles have sides in the ratio 3:5, their areas are in the ratio 9:25. If one area is 36 cm², the other becomes 100 cm² when the smaller triangle matches the 3:5 scale. Questions may give side ratios and ask for area ratio, or give area ratio and ask for side ratio. Using 3:5 as the area ratio directly and ignoring the square relationship.
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