# Perimeter Ratios in Similar Polygons

When it comes to similar polygons, the perimeter ratio matches the ratio of any pair of corresponding sides. $$ P : P' = l : l' $$

In layman's terms, if two polygons are similar, their perimeters reflect their scale factor, k.

This rule highlights that the similarity in shape carries over to the perimeters as well.

This theorem, however, __applies strictly to similar polygons__.

## A Concrete Example

Consider two similar triangles, though the concept is applicable across all sorts of polygons, whether they’re regular or not.

Here, the scale factor is simply k=2.

$$ \frac{6}{3} = \frac{4}{2} = \frac{7.2}{3.6} = 2 $$

The same goes for the perimeter ratio of these polygons, which is also 2.

$$ \frac{6+4+7.2}{3+2+3.6} = \frac{17.2}{8.6} = 2 $$

Thus, the perimeter ratio in similar polygons matches the scale factor of their corresponding sides.

## The Proof

Take two similar polygons for example.

Similarity by definition means the sides correspond and are proportional.

$$ a : a' = b : b' = c : c' $$

This indicates a constant ratio across corresponding sides, equal to k:

$$ \frac{ a }{ a' } = \frac{ b }{ b' } = \frac{ c }{ c' } = k $$

With this consistency, it can also be expressed as:

$$ \frac{ a+b+c }{ a'+b'+c' } = k $$

Where the sum a+b+c=P is the first polygon's perimeter, and a'+b'+c'=P' represents the second's.

$$ \frac{ P }{ P' } = k $$

Knowing that the ratio between any two corresponding sides equals k

$$ \frac{ a }{ a' } = k $$

it follows their perimeters show the same scale factor.

$$ \frac{ P }{ P' } = \frac{ a }{ a' } = k $$

This confirms the perimeter ratio is directly tied to the ratio between corresponding sides of similar polygons.

$$ P : P' = a : a' $$

And there you have it.