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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 12cm2

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 12cm2 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 60cm3. When the lengths are doubled, its volume increases to 6 x 8 x 10 = 480cm3

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:     x2:y2

=Volume ratio:     x3:y3

 

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