# Similar Triangles and Polygons

Polygons are 'similar' if they are exactly the same shape, but can be different sizes.

Try this Drag any orange dot at R. The polygon will remain similar to the polygon LMNO even though it changes size.

Similar polygons have the same shape, but can be different sizes. Specifically, two polygons are
similar if two things are true:

1. The corresponding sides of each are in the same proportion
2. The corresponding interior angles are the same (congruent).

In the figure above, click 'reset'. The polygon PQRS is similar to LMNO. Each side of PQRS is exactly twice the length of the corresponding side in LMNO. So, for example, QR is twice MN and so on.

In addition, each interior angle of PQRS is congruent to the corresponding angle in LMNO.
For example the angles P and L are congruent.

## Formal notation

Given two similar polygons ABCD and JKLM, we can write
ABCD ~ JKLM
which is read as "polygon ABCD is similar to polygon JKLM". The wavy line symbol means 'similar to'.

## Rotation and reflection

Polygons can still be similar even if one of them is rotated, and/or mirror image of the other. In the figure below, all three polygons are similar.

Starting with the polygon on the left, the center polygon is rotated clockwise 90°, the right one is flipped vertically. This is illustrated in more depth in "Similar Triangles", but is true for all similar polygons, not just triangles.

In the figure at the top of the page, click 'reflect'. The polygon PQRS is now reflected over the vertical green line. It is still similar to LMNO. The trick is to notice which side corresponds to which. In this case the side QP corresponds to NO for example. And the angle P corresponds to the angle O.

A way to remember these transformations is to imagine PQRS cut out of cardboard. No matter how you rotate it or turn it over, it will remain similar to LMNO.

## Properties of Similar Polygons

1. ### Corresponding angles are the same

In the figure at the top, the following angles are congruent:
P=L,   S=O,   R=N   and Q=M
From this, it follows that the corresponding exterior angles will also be the same.
2. ### Corresponding sides are all in the same proportion

By definition each pair of corresponding sides are in the same proportion, or ratio.
Formally, in two similar polygons PQRS and LMNO :
3. ### Corresponding diagonals are in the same proportion

In each polygon the corresponding diagonals are in the same proportion. Their ratio is the same as the ratio of the sides.
4. ### Area ratio

The ratio of the areas of the two polygons is the square of the ratio of the sides. So if the sides are in the ratio 3:1 then the areas will be in the ratio 9:1. This is illustrated in more depth for triangles in "Similar Triangles - ratio of areas", but is true for all similar polygons, not just triangles.

## Regular Polygons

For any two regular polygons with the same number of sides:
1. They are always similar. Since they have the sides all the same length they must always be in the same proportions, and their interior angles are always the same, and so are always similar.
2. The apothems and radii are in the same proportions as each other and the sides.

## Things to try

In the figure at the top of the page, assume that we have proved that PQRS and LMNO are similar polygons.
1. Click on 'reset' and 'hide details'.
2. Drag the point R to resize PQRS to any new size.
3. Calculate the angles P and S, and the length of the side PS