Tutorial 1.4: Part 4

Derived Units

How do we record measurements that are not directly measurable in terms of base units? For example, what do we record as the unit for area? We record area in terms of standard squares.

Derived Units

For example, the figure shown has an area equivalent to 4 meter squares and is written as 4.0 m2

How many decimeter squares would be needed to cover the same area?

Since there are 10 decimeters in 1 meter, we note that 10 decimeter squares would be needed to cover the first row of a meter square. We will need ten such rows to cover the entire grey square.

Since 100 dm2 are equivalent to 1 m2, it takes 400 dm2 to cover the enitire 4 meter squares.

Here is a way to write out the same thing using conversion factors: $$ 4 \, m^2 \times \frac{10^2 \,dm^2}{1 \,m^2} = 400 \, dm^2 $$

From this diagram we can see that 100 cm2 are equivalent to 1 dm2, so 100 x 100 cm2 are needed to cover 1 meter square.

Practicing Area Conversions

Factor Number Name Symbol
1012 1 000 000 000 000 tera T
109 1 000 000 000 giga G
106 1 000 000 mega M
103 1000 kilo k
102 100 hecto h
101 10 deka da

Factor Number Name Symbol
10-1 0.1 deci d
10-2 0.01 centi c
10-3 0.001 milli m
10-6 0.000 001 micro µ
10-9 0.000 000 001 nano n
10-12 0.000 000 000 001 pico p

How many dm3 are equivalent to 1 m3?

10 x 10 are needed to cover the "floor", and the cube contains 10 "floors", so 103 are needed.

How many cm3 are equivalent to 1 dm3 ? By definition 1 dm3 = 1 L. How many mL = 1 L?

Again, 10 x 10 are needed to cover the "floor", and the cube contains 10 "floors", so 103 are needed.

Note: 1 L = 1 dm3; 1 L = 1000 mL ; 1 dm3 = 1000 cm3; 1 mL = 1 cm3 .

How many cm3 are equivalent to 1 m3?

1000 cm3 are equivalent to 1 dm3 and 1000 dm3 are equivalent to 1 m3, so 1000 x 1000 (1 million) cm3 are equivalent to 1 m3 ?

Practicing Volume Conversions

Factor Number Name Symbol
1012 1 000 000 000 000 tera T
109 1 000 000 000 giga G
106 1 000 000 mega M
103 1000 kilo k
102 100 hecto h
101 10 deka da

Factor Number Name Symbol
10-1 0.1 deci d
10-2 0.01 centi c
10-3 0.001 milli m
10-6 0.000 001 micro µ
10-9 0.000 000 001 nano n
10-12 0.000 000 000 001 pico p