Finding the ratio of areas or volumes given the length of a side of a 2 or 3 dimensional figure was always a time consuming task. With the help of the knowledge you are going to acquire now, this will be a simple and quick task.

*In any two dimensional figure, if the corresponding sides are in the ratio a:b, then their areas are in the ratio a ^{2}:b^{2 }*

Two dimensional figures can be any polygon like square, rectangle, rhombus, trapezium, hexagon, etc. It can also be a triangle or a circle. The sides, referred in the statement above, can be length, breadth or even diagonal in case of a polygon. In case of a circle the sides will be represented by radius or diameter or circumference. In triangle it can be sides or height of a triangle.

To understand the above concept let us take few examples:

**Problem: The sides of a hexagon are enlarged by three times. Find the ratio of the areas of the new and old hexagon.**

To solve this problem students normally assume the sides of smaller hexagon as x. Hence the corresponding sides of the enlarged hexagon become 3x. Then they calculate areas of respective hexagon using formula, Area = (3?3*x^{2})/2. Then they will find the ratio of areas of both the hexagons.

Using the shortcut above, we know ratio of the corresponding sides of the two hexagons is a:b = 1:3

Therefore, ratio of their areas is given by a^{2}:b^{2} = 1^{2}:3^{2} = 1:9

**Problem: The ratio of the diagonal of two squares is 2:1. Find the ratio of their areas. **

Using the above shortcut, the ratio of their area will be 2^{2}:1^{2} = 4:1

**Problem: The ratio of the radius (or diameter or circumference) of two circles is 5:7. Find the ratio of their areas.**

Using the above shortcut, the ratio of their area will be 5^{2}:7^{2} = 25:49

The same logic can be extended to any 3-dimensional figure like cube or sphere or cone or any other.

*In any two 3-dimensional figures, if the corresponding sides or other measuring lengths are in the ratio a:b, then their surface area are in the ratio a ^{2}:b^{2} andvolumes are in the ratio a^{3}:b^{3}*

**Problem: The sides of two cubes are in the ratio 2:1, find the ratio of their surface area and volumes.**

The ratio of their surface area is 2^{2}:1^{2} = 4:1 and ratio of volumes would be 2^{3}:1^{3} = 8:1

**Problem: The radiuses of two spheres are in the ratio 3:4, find the ratio of their surface area and volumes.**

The ratio of their surface area is 3^{2}:4^{2} = 9:16 and ratio of volumes would be 3^{3}:4^{3} = 27:64

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In the second example the ratio 22:12 does not give 4:1 please check again.

if rhe ratio is thrice .. ?

Yeah……its excellent….itz a gr8 help….4 me………

yeah grace ur rite m also having s same problem i didnt get d shortcut method for division

This is excellent. I am trying to prepare for GMAT and this kind of concepts I was looking out for. It is a great help. Thanks again.

Hi Vineet, I came across your post on the shortcut for two divisor division. I am still struggling with it. Do you happen to have a video for it or a YouTube Channel? If not, do offer online private lessons? Thanks!