Similar Triangles: Finding Area Ratio from 3:4 Length Ratio

Q: Why do I need to square the length ratio to get the area ratio?

Because area is two-dimensional! When you scale a shape, both length and width change. If each dimension changes by factor 3/4, then area changes by $ \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} $.

Q: Does this work for all similar shapes, not just triangles?

Yes! This area-to-length relationship applies to all similar shapes - triangles, rectangles, circles, etc. The area ratio is always the square of the length ratio.

Q: What if the problem gave me the area ratio first?

Then you'd take the square root to find the length ratio! If area ratio is 9:16, then length ratio is $ \sqrt{\frac{9}{16}} = \frac{3}{4} $.

Q: Can I use this method with decimal ratios too?

Absolutely! If length ratio is 0.5:1, then area ratio is $ (0.5)^2 = 0.25 $, or 0.25:1. The squaring rule works with any number format.

Area Ratios with Squared Length Proportions

Here are two similar triangles. The ratio of the lengths of the sides of the triangle is 3:4, what is the ratio of the areas of the triangles?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the ratio of triangle areas

00:03 Let's mark the triangles as 1,2

00:07 Find the similarity ratio

00:15 The area ratio equals the similarity ratio squared

00:24 Make sure to square both numerator and denominator

00:35 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Here are two similar triangles. The ratio of the lengths of the sides of the triangle is 3:4, what is the ratio of the areas of the triangles?

Step-by-step solution

Let's call the small triangle A and the large triangle B, let's write the ratio:

$\frac{A}{B}=\frac{3}{4}$

Square it:

$\frac{S_A}{S_B}=(\frac{3}{4})^2$

$\frac{S_A}{S_B}=\frac{9}{16}$

Therefore, the ratio is 9:16

Final Answer

9:16

Key Points to Remember

Essential concepts to master this topic

Area Rule: Area ratio equals the square of length ratio
Technique: Square both sides: $(\frac{3}{4})^2 = \frac{9}{16}$
Check: Verify with actual measurements: smaller area is 9/16 of larger ✓

Common Mistakes

Avoid these frequent errors

Using the same ratio for both length and area
Don't assume area ratio = length ratio = 3:4! This gives completely wrong answers because area is two-dimensional. Always square the length ratio to get area ratio: $(\frac{3}{4})^2 = \frac{9}{16}$ .

Practice Quiz

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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 need to square the length ratio to get the area ratio?

Because area is two-dimensional! When you scale a shape, both length and width change. If each dimension changes by factor 3/4, then area changes by $\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$ .

Does this work for all similar shapes, not just triangles?

Yes! This area-to-length relationship applies to all similar shapes - triangles, rectangles, circles, etc. The area ratio is always the square of the length ratio.

What if the problem gave me the area ratio first?

Then you'd take the square root to find the length ratio! If area ratio is 9:16, then length ratio is $\sqrt{\frac{9}{16}} = \frac{3}{4}$ .

How can I remember this rule?

Think of a simple example: if you double all lengths (ratio 1:2), the area becomes 4 times bigger (ratio 1:4). Length squared = area!

Can I use this method with decimal ratios too?

Absolutely! If length ratio is 0.5:1, then area ratio is $(0.5)^2 = 0.25$ , or 0.25:1. The squaring rule works with any number format.

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Similar Triangles: Finding Area Ratio from 3:4 Length Ratio

Area Ratios with Squared Length Proportions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to square the length ratio to get the area ratio?

Does this work for all similar shapes, not just triangles?

What if the problem gave me the area ratio first?

How can I remember this rule?

Can I use this method with decimal ratios too?

🌟 Unlock Your Math Potential

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