10.20 Similar areas and volumes (Extended)

All pairs of corresponding sides in similar figures are in the same ratio, and we can use this ratio to find the scale factor.

Since this scale factor affects the length of sides, it is also called the length scale factor or linear scale factor.

But what happens to the area of a figure when we enlarge it by a length scale factor? Does it also enlarge? Is it by the same or some other related scale factor?

Suppose we have a square with side lengths $2$2 cm and we enlarge it by a length scale factor of $3$3.

Our lengths have been scaled by a factor $3$3, but our area has gone from $4$4 cm² to $36$36 cm². It has been scaled by a factor of $9$9!

Areas of similar figures do not scale by the same factor as the linear scale factor, and for this reason, they have their own scale factor called the area scale factor.

So, is there any way to predict what the area scale factor of a shape will be if we know its length scale factor? The answer is yes, and we'll see why.

 

Area scale factor

Let's say we have a square of side length $a$a units. This square would therefore have area $a^2$a2 units².

Let's say we want to scale it by a length scale factor of $k>1$k>1 (this value can actually be less than one, in which case the square would shrink).

What is the area now?

The side length of our new square is $a\times k$a×k units. We can use this to figure out the new area.

New Area	$=$=	$\left(a\times k\right)\times\left(a\times k\right)$(a×k)×(a×k)	units2
	$=$=	$\left(a\times k\right)^2$(a×k)2	units2
	$=$=	$a^2\times k^2$a2×k2	units2

So, if our old area was $a^2$a2 units² and our new area is $a^2\times k^2$a2×k2 units², we have scaled our area by a factor of $k^2$k2. Since this works for any side length $a$a units and any scale factor $k$k, we know that if we scale any square by length scale factor $k$k, the area scale factor will always be $k^2$k2.

 

Remember back to when you were first learning about areas. Area is defined as the amount of unit squares that fit into a two dimensional shape. 

So, if we have a shape whose area is $18.79$18.79 cm², what we really mean is that it can fit $18.79$18.79 lots of $1$1 cm² squares inside it.

If we enlarge or shrink any figure, no matter how irregular, by a length scale factor of $k$k, the areas of all these squares inside it will each scale by a factor of $k^2$k2. 

Hence, the total area sum will also scale by a factor of $k^2$k2! This means that for absolutely any figure that is scaled by a length scale factor of $k$k, the area will always scale by an area scale factor of $k^2$k2.

 

Practice question

Question 3

 

Volume scale factor

Given what you've learnt above, can you guess how the volume scale factor of a three dimensional figure might change depending on the length scale factor $k$k? If your answer is that it will be $k^3$k3, you're correct!

Given a cube of side length $a$a units, its volume would be $a^3$a3 units³.

 

Again, let's say we want to scale it by a length scale factor of $k>1$k>1.

The side length of our new cube is $a\times k$a×k units. We can again use this to figure out the new volume.

New Area	$=$=	$\left(a\times k\right)\times\left(a\times k\right)\times\left(a\times k\right)$(a×k)×(a×k)×(a×k)	units3
	$=$=	$\left(a\times k\right)^3$(a×k)3	units3
	$=$=	$a^3\times k^3$a3×k3	units3

Hence, we know that if we scale any cube by length scale factor $k$k, the volume scale factor will always be $k^3$k3.

 

The same principal applies here as before. Volumes are just measured in unit cubes. If we have a three dimensional figure whose volume is $253.79$253.79 cm³, what we really mean is that it can fit $253.79$253.79 lots of $1$1 cm³ cubes inside it.

If we enlarge or shrink any figure, no matter how irregular, by a length scale factor of $k$k, the volumes of all these cubes inside it will each scale by a factor of $k^3$k3. 

Hence, the total volume sum will also scale by a factor of $k^3$k3! Again, this means that for absolutely any figure that is scaled by a length scale factor of $k$k, the volume will always scale by an volume scale factor of $k^3$k3.

 

 

Practice question

Question 4

 

Lines and similarity

As it turns out, when two figures are similar, it's not just the side lengths that are in proportion, it's absolutely any pair of corresponding lines between the shapes. This could include radii, diameters or diagonals. This is true for three dimensional shapes as well.

This is why the length scale factor is often called the linear scale factor. The linear scale factor can be found from the ratio of absolutely any pair of corresponding lines in the figure.

 

Practice question

Question 5

 

Surface area scale factor

Given what we've just learned about the linear scale factor, we can find the relationship between the linear scale factor and surface area scale factor of a solid.

Similar squares in three dimensional space with a length scale factor $k$k will have an area scale factor $k^2$k2, given what we know about areas of similar figures.

Given that the surface area of a shape is just the area sum of all the little squares that fit on its surface, whenever we scale a solid using a linear scale factor, all the little square side lengths will scale also. 

Hence, the total surface area sum will just scale by $k^2$k2, as was the case in two dimensions!

 

Practice question

Question 6