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# Similar Figures

A model airplane may come in a range of different "scales," and these scales are written as ratios. Common model airplane scales include $\frac{1}{48}$ and $\frac{1}{72}$. This means that the airplane is exactly $\frac{1}{48}$ or $\frac{1}{72}$ the size of the real plane in each dimension.

The same concept (known as "scale factor") can be applied to much simpler, two-dimensional geometric shapes. This concept is also related to something called "similarity." But what exactly is similarity? How do we determine similarity? And perhaps most importantly, what can it teach us about math? Let's find out:

## The definition of similarity

If we look at two figures and see the following characteristics, we know that the two figures are similar:

• They have the same shape
• Their corresponding angles are congruent
• The ratios of the lengths of their corresponding sides are equal

Before we get any further, let's define a few of these terms:

Congruence is essentially the same thing as equality. For example, if two angles both have 33 degrees, we can call them congruent.

Corresponding angles and sides are angles and sides that are in the same relative position.

## Examples of similar figures

Let's look at a few examples of similar figures:

Here we see two pentagons. Their angles are corresponding and congruent, and so are their sides. The sides also have ratios of the same lengths. Because of this, we can safely say that $\mathrm{ABCDE}\sim \mathrm{VWXYZ}$.

A few important notes: The symbol "∼" means "is similar to." You should also know that the order of the letters matters. While ABCDE may be similar to VWXYZ, it is not similar to VZYXW.

Here's another example of similarity:

We are told the cylinders are similar and want to know the radius of the smaller one.

First, let's find the scale factor. One cylinder is three times as high as the other cylinder. This means that there's a scale factor of $\frac{1}{3}$.

Next, we need to find the radius of the smaller cylinder. All we need to do here is divide 1.8 by 3.

$\frac{1.8}{3}=0.6$

We know both the radius and the height of the larger cylinder are exactly three times greater than those of the smaller cylinder. This means that these two figures are similar and therefore the smaller cylinder has a radius of 0.6 cm.

Here's another example:

As we can see, even if we flip the figure horizontally, translate it down, and dilate it by a scale factor of $\frac{1}{2}$ this still leaves us with a smaller yet similar figure. All four of these hexagons are similar.

Here's another example:

If we rotate this pentagon by 180 degrees, we get something like this:

Now we can dilate it about the origin by a scale factor of $\frac{2}{1}$. This operation leaves us with something like this:

Throughout all of these transformations, the various pentagons remained similar.

## Topics related to the Similar Figures

Side-Angle-Side Similarity

Squares Circumscribed by Circles

Side-Side-Side Similarity

## Flashcards covering the Similar Figures

Common Core: High School - Geometry Flashcards

## Practice tests covering the Similar Figures

Common Core: High School - Geometry Diagnostic Tests