Similarity

Many words have a meaning in the English language and then a different meaning in mathematics. Similar is one of these words. In mathematics it does not mean that something is like something else. It has a very specific meaning.

Two shapes are similar if the sides are in proportion.

So if one side is twice the size of its corresponding side in the other shape, then all of the sides must be twice the size of their corresponding sides.

With triangles, it is sufficient to know that all three angles in the triangle are equal to the corresponding angle in a different shape to know that the two shapes are similar.

So if we know that all of the angles are equal, then we know the shape is similar and then we can work out missing lengths, as the lengths must all be in the same proportion.

Worked Examples

Exercise

Let’s practice this in exercise 7 from page 148 to 149 of the extended textbook:

The answers to this exercise are detailed below:

Area and volume of similar shapes

If the length of a similar shape is “k times” bigger, then the area of that shape will be “k² times” bigger (i.e. if the length is 5 times bigger, then the area will be 25 times bigger). This is because the area has 2 dimensions, which are effectively multiplied together.

A comparable thing happens with volume, which we shall discuss further below.

Example

Example 2

Exercise

Now let’s try questions 1 to 12 from exercise 8 on page 151 of the extended textbook:

The answers are below:

Volume of similar objects

The situation with volume is comparable to the one with area.

If the length of a similar shape is “k times” bigger, then the area of that shape will be “k³ times” bigger (i.e. if the length is 5 times bigger, then the area will be 125 times bigger). This is because the area has 3 dimensions, which are effectively multiplied together.

Example 1

Example 2

Exercise

Now let’s complete exercise 9 from pages 154 to 156 of the extended textbook:

Below are the answers:

Congruence

Congruence is a stricter condition than similarity, which basically means that two shapes have exactly the same shape and exactly the same size. We are interested in what conditions are sufficient to be sure that two shapes are congruent.

With triangles we have 4 possible criteria which if they are met show that two shapes are congruent. We remember them using the acronyms: SSS, SAS, ASA, AAS and RHS.

Let’s try the questions below (not in the textbook):

The answers are below: