Scale Factors - I

or 'Why There Are No Giant Spiders!

Giant creatures the stuff of many science fiction movies ... from giant ants and spiders to 10-storey tall babies. But is it really possible for creatures to be so large? If so, why aren't there any 6 metre tall spiders? On this page we'll try to explain why ants and spiders could never be as big as an elephant ... and still look the same. The reason for this has to do with scale factors ...

Here's a simple box. Beside it is one twice as big.
You can tell it's twice as big because we've put a scale beside it.

Actually, to be more precise, the second box is twice as long as the first one.
The correct way to say this is that it has been scaled up by a factor of two.

By making the box twice as long, we've also doubled the width and height.

Here's the same diagram, only this time we've shaded one face. The second box is still twice as long as the first one. Notice what happens to the area of one face of the box when it's twice as long ... The AREA has been increased by a factor of FOUR.

Here's the same diagram again, only this time we've shaded all the boxes. The second box is still twice as long as the first one. Notice what happens to the volume of the box when it's twice as long ... The VOLUME has been increased by a factor of EIGHT.

Increasing the length of an object by a factor of 2 increases the area by a factor of 4 and the volume by a factor of 8.


Let's start over with another set of boxes.



Here's a simple box. Beside it is one three times as big.
You can tell it's three times as big because we've put a scale beside it.

To be more precise, the second box is three times as long as the first one.
The correct way to say this is that it has been scaled up by a factor of three.

By making the box three times as long, we've also tripled the width and height.

Here's the same diagram, only this time we've shaded one face. The second box is still three times as long as the first one. Notice what happens to the area of one face of the box when it's three times as long ... The AREA has been increased by a factor of NINE.

Here's the same diagram again, only this time we've shaded all the boxes. The second box is still three times as long as the first one. Notice what happens to the volume of the box when it's three times as long ... The VOLUME has been increased by a factor of TWENTY-SEVEN.

Increasing the length of an object by a factor of 3 increases the area by a factor of 9 and the volume by a factor of 27.


Do you see the pattern? Let's summarize what happened (and add a few more examples) in a table:

	LENGTH	AREA	VOLUME
Scale Factor 2	2 times	4 times	8 times
Scale Factor 3	3 times	9 times	27 times
Scale Factor 4	4 times	16 times	64 times
Scale Factor 5	5 times	25 times	125 times
Scale Factor X	X times	X2 times	X3 times

Increasing the length of an object by factor X increases the area by factor X² and the volume by factor X³