1. Mathematically Similar
Created by
Bernard Lafferty
Bsc(Hons) Mathematics GIMA
15-Nov-18

2. Mathematically Similar
An object is said to be mathematically similar to another
if the only difference between them is a scaling (k) in
ALL directions
The objects can be complex in shape and be in one, two or
three dimensions. I will use a square shape to show how the
scaling factor is related to the length, area and volume of
the object in different dimensions.
15-Nov-18

3. Mathematically Similar
Consider the simple case of one dimension.
Draw a line 2 unit long and then draw another 4 units long.
x direction
2 units long
4 units long
It should be quite clear that second line is twice the first.
Hence the scaling factor is k = 2.
15-Nov-18

4. Mathematically Similar
Consider the case in two dimensions.
Draw a square with sides 2 units long and then draw another 4 units long. x y
Area = 2x2 = 4 x y
Sides 2 units long
Area = 4x4 = 16
Sides 4 units long
It should be quite clear that second area is four times the first.
Hence the scaling factor for two dimension is k2.
In our example we have this case 22 = 4.
15-Nov-18

5. Mathematically Similar
Consider the case in three dimensions.
Draw a cube with sides 2 units long and then draw another 4 units long. x y z
Volume = 2x2x2 = 8 x y z
Sides 2 units long
Volume = 4x4x4 = 64
Sides 4 units long
It should be quite clear that Volume area is eight times the first.
Hence the scaling factor for two dimension is k3.
In our example we have this case 23 = 4.
15-Nov-18

6. Summary One dimension : Length is changed by a factor of k
If we scale a simple or complex object by a factor of k units
it has the effect of the following :-
YOU NEED TO REMEMBER THE FOLLOWING
One dimension : Length is changed by a factor of k
Two dimensions : Area is changed by a factor of k2
Three dimensions : Volume is changed by a factor of k3
Note : If k < 1 then the scaling is reduced in size.
If k > 1 then the scaling is increased in size.
15-Nov-18

7. Example Q1. An A4 sheet of paper has area 600mmm2. If it is cut
in half both long and short ways what is the value of area left.
Solution
Scaling factor is k = 1/2
New area is A=(1/2)2 x 600 = (1/4) x 600 = 150mm2
15-Nov-18

8. Example
Q2. An small cereal box contains 100g of cereal. How much
does a large box contain if it is double the size. Assume the
objects are mathematically similar.
Solution
Scaling factor is k = 2
New Volume is V=(2)3 x 100 = (8) x 100 = 800g.
THE END
15-Nov-18