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 m^{2}

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 dm^{2} are equivalent to 1 m^{2}, it takes 400 dm^{2} 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 cm^{2} are equivalent to 1 dm^{2}, so 100 x 100 cm^{2} are needed to cover 1 meter square.

Practicing Area Conversions

Factor | Number | Name | Symbol |
---|---|---|---|

10^{12} |
1 000 000 000 000 | tera | T |

10^{9} |
1 000 000 000 | giga | G |

10^{6} |
1 000 000 | mega | M |

10^{3} |
1000 | kilo | k |

10^{2} |
100 | hecto | h |

10^{1} |
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 dm^{3} are equivalent to 1 m^{3}?

10 x 10 are needed to cover the "floor", and the cube contains 10 "floors", so 10^{3} are needed.

How many cm^{3} are equivalent to 1 dm^{3} ? By definition 1 dm^{3} = 1 L. How many mL = 1 L?

Again, 10 x 10 are needed to cover the "floor", and the cube contains 10 "floors", so 10^{3} are needed.

Note: 1 L = 1 dm^{3}; 1 L = 1000 mL ; 1 dm^{3} = 1000 cm^{3}; 1 mL = 1 cm^{3} .

How many cm^{3} are equivalent to 1 m^{3}?

1000 cm^{3} are equivalent to 1 dm^{3} and 1000 dm^{3} are equivalent to 1 m^{3}, so 1000 x 1000 (1 million) cm^{3} are equivalent to 1 m^{3} ?

Practicing Volume Conversions

Factor | Number | Name | Symbol |
---|---|---|---|

10^{12} |
1 000 000 000 000 | tera | T |

10^{9} |
1 000 000 000 | giga | G |

10^{6} |
1 000 000 | mega | M |

10^{3} |
1000 | kilo | k |

10^{2} |
100 | hecto | h |

10^{1} |
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 |