__
Similar Solids__

Similar shapes are shapes with the
__same angles __
and the same __proportion of
side lengths__.

Therefore, two triangles are similar if...

The
angles in both shapes are identical

The
sides are all in the same ratio

Sides are in the same ratio and the angle __between them__ (included
angle) is the same

__Other shapes__

Below is a rectangle of sides 3x4

It has an area of 12cm^{2}

If I now enlarge all of its lengths by a
scale factor 2 (double them), the new lengths become 6cm and 8cm.

You might expect the new area to be double
the original 12cm^{2} but in fact it is now 6 x 8 = 48. The sides have
each doubled once but the area has doubled twice (multiplied by 4)

Now picture a cuboid of lengths 3 x 4 x 5.
Its volume is 60cm^{3}. When the lengths are doubled, its volume
increases to 6 x 8 x 10 = 480cm^{3}

When the lengths of a 3D object double,
its volume doubles three times (which is the same as multiplying by 8). So far
we have only dealt with doubling. Below are rules for dealing with any
situation. It will even work for scale factors like 1.56 etc!

To get from a side ratio to an area ratio,
square the ratio. Eg) a side ratio of 3:4 is an area ratio of 9:16

To get from a side ratio to a volume
ratio, cube the ratio. Eg) a side ratio of 3:4 is an area ratio of 27:64

To get from an area ratio to a volume
ratio, change it into a side ratio first by square rooting, then cube the ratio.
Eg) an area ratio of 25:49 is a side ratio of 5:7 and hence a volume ratio of
125 : 343

__Summary__

Side ratio:
x:y

=Area ratio:
x^{2}:y^{2}

=Volume ratio:
x^{3}:y^{3}