Area and Similar triangles. How to find the ratio of areas from the similarity ratio. All you have to do is...

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

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

Triangle 2

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

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

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

What about the perimeter of similar triangles?

If 2 triangles are similar, their perimeters have the exact same ratio

For instance if the similarity ratio of 2 triangles is $$\frac 3 4 $$ , then their perimeters have a ratio of $$\frac 3 4 $$

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

Example 2

Triangle 1

Perimeter of Triangle #1

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

Triangle 2

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} $

Relationship of Area/Perimeter for Similar Triangles

Similar Triangles Area and Perimeter Relationship Demo

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 $