Similar Triangles: Finding Area Ratio from 9:8 Length Ratio

Area Ratios with Side Length Ratios

The triangle ABC is similar to the triangle DEF.

The ratio between the lengths of their sides is 9:8.

What is the ratio between the areas of the triangles?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:11 Let's find the ratio of the triangle areas.
00:15 These triangles are similar. So, we use the similarity ratio from the given data.
00:21 Remember, the area ratio equals the square of the similarity ratio.
00:26 Now, let's substitute the similarity value and calculate the area ratio.
00:32 Don't forget to square both the top and bottom numbers.
00:40 And there you go! We've found the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

The triangle ABC is similar to the triangle DEF.

The ratio between the lengths of their sides is 9:8.

What is the ratio between the areas of the triangles?

2

Step-by-step solution

We multiply the ratio by 2

9:8=18:16 9:8=18:16

Raised to the power of 2:

92:82=81:64 9^2:8^2=81:64

3

Final Answer

81:64

Key Points to Remember

Essential concepts to master this topic
  • Rule: Area ratio equals side ratio squared for similar triangles
  • Technique: From 9:8 side ratio, calculate 92:82=81:64 9^2:8^2 = 81:64
  • Check: Verify 81÷64 = 1.265625 matches (9÷8)² = 1.265625 ✓

Common Mistakes

Avoid these frequent errors
  • Using the side ratio directly as the area ratio
    Don't use 9:8 as the area ratio = completely wrong answer! Areas scale with the square of linear dimensions, not linearly. Always square the side ratio to get the area ratio.

Practice Quiz

Test your knowledge with interactive questions

If it is known that both triangles are equilateral, are they therefore similar?

FAQ

Everything you need to know about this question

Why do I square the side ratio to get the area ratio?

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Because area is a 2D measurement! When you scale a triangle by factor 9:8, both length AND width scale by that ratio, so area scales by (9:8)×(9:8)=92:82 (9:8) \times (9:8) = 9^2:8^2 .

What if the problem asked for perimeter ratio instead?

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Perimeter ratio would be the same as the side ratio (9:8) because perimeter is a 1D measurement. Only areas and volumes get squared or cubed!

How can I remember when to square the ratio?

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Think dimensions: 1D measurements (length, perimeter) keep the original ratio. 2D measurements (area) square it. 3D measurements (volume) cube it.

Is 81:64 the same as writing it as a decimal?

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Yes! 8164=1.265625 \frac{81}{64} = 1.265625 as a decimal. Both forms are correct, but ratios like 81:64 often show the relationship more clearly in geometry problems.

What if I got 18:16 by doubling both numbers?

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That's not how ratios work! 18:16 simplifies to 9:8, which is the side ratio, not the area ratio. You need to square each number: 92:82 9^2:8^2 .

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