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 a2:b2
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*x2)/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 a2:b2 = 12:32 = 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 22:12 = 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 52:72 = 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 a2:b2 andvolumes are in the ratio a3:b3
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 22:12 = 4:1 and ratio of volumes would be 23:13 = 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 32:42 = 9:16 and ratio of volumes would be 33:43 = 27:64