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Area and Perimeter of Similar Triangles

What is true about the ratio of the area of similar triangles?

Answer: If 2 triangles are similar, their areas are the square of that similarity ratio (scale factor)

For instance if the similarity ratio of 2 triangles is $$\frac 3 4 $$ , then their areas have a ratio of $$\frac {3^2}{ 4^2} = \frac {9}{16} $$

Let's look at the two similar triangles below to see this rule in action.

Example 1

It's easiest to see that this is true if you look at some specific examples of real similar triangles.

Triangle 1
area

$$ Area = \frac{1}{2}\cdot{12}\cdot{4} \\ Area = 24 $$

Triangle 2
area

$$ Area = \frac{1}{2}\cdot{24}\cdot{8} \\ Area = 96 $$

Notice: $$ \frac{24}{96} = \frac{1}{4} $$
area

Therefore, if you know the similarity ratio, all that you have to do is square it to determine ratio of the triangle's areas.

Example 2
Triangle 1
perimeter Perimeter of Triangle #1

Perimeter $$ = 6 + 8 + 10 = 24 $$

Triangle 2
perimeter Perimeter of Triangle #2

Perimeter $$ = 5 + 3 +4 = 12 $$


The ratio of the perimeter's is exactly the same as the similarity ratio!

$ \frac{\text{perimeter #1}}{\text{perimeter #2}} = \frac{24}{12} = \frac{2}{1} $

Practice Problems

Problem 1

$$\triangle ABC$$ ~ $$\triangle XYZ$$ and have a scale factor (or similarity ratio) of $$ \frac{3}{2} $$.

What is the ratio of their areas?

$ \text{Ratio of areas} = (\text{similarity ratio})^2 \\ = \Big(\frac{3}{2}\Big)^2 \\ = \frac{9}{4} $

Problem 2

$$\triangle ABC$$ ~ $$\triangle XYZ$$. The ratio of their perimeters is $$ \frac{11}{5} $$, what is their similarity ratio and the ratio of their areas?

What is the ratio of their areas?

$ \text{ratio of perimeters} = \text{similarity ratio} \\ \text{similarity ratio} = \frac{11}{5} \\ \text{ratio of areas} = (\text{similarity ratio})^2 \\ = \Big(\frac{11}{5}\Big)^2 \\ \text{ratio of areas} = \frac{121}{25} $

Problem 3

$$\triangle ABC$$ ~ $$\triangle XYZ$$. The ratio of their areas is $$ \frac{36}{17} $$, what is their similarity ratio and the ratio of their perimeters?

What is the ratio of their areas?

$ \text{ratio of areas} = (\text{similarity ratio})^2 \\ (\text{similarity ratio})^2 = \text{ratio of areas} \\ \text{similarity ratio} = \sqrt{\text{ratio of areas} } \\ = \sqrt{\Big(\frac{36}{17} \Big) } \\ = \frac{\sqrt{36}}{\sqrt{17 } } \\ = \frac {6 }{\sqrt{17 } } $

Problem 4

$$\triangle HIJ$$ ~ $$\triangle XYZ$$. The ratio of their areas is $$ \frac{25}{16}$$, if XY has a length of 40, what is the length of HI?

What is the ratio of their areas?

We need to find the similarity ratio first, since that ratio gives us a proportion between corresponding sides.

$ \text{ratio of areas} = (\text{similarity ratio})^2 \\ (\text{similarity ratio})^2 = \text{ratio of areas} \\ \text{similarity ratio} = \sqrt{\text{ratio of areas} } \\ = \sqrt{\Big(\frac{25}{16} \Big) } \\ \text{similarity ratio} = \frac{5}{4 } $

Now, that you have found the similarity ratio, you can set up a proportion to solve for HI

$ \frac{5}{4 } = \frac{HI}{XY} \\ \frac{5}{4 } = \frac{HI}{40} \\ \frac{40 \cdot 5}{4 } = HI \\ HI = 50 $


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