# Decimal metric system: length, mass, capacity, surface and volume

To make measurements, you need a system of units, that is, a set of magnitudes with which to compare those things that you want to measure.

The decimal metric system is a system of units in which the multiples and sub-multiples of the unit of measurement are interrelated by multiples or sub-multiples of $10$.

For example, the following belong to metric units: gram and kilogram (to measure the mass), meter and centimeter (for measuring length) or liter (to measure capacity).

Apart from the metric system, there are other systems of units: the Anglo-Saxon system, the so-called traditional measurements, etc.

## Measurements of length

The unit for measuring length is the meter. However, there are other units:

Name Symbol Equivalence
kilometer km 1000 m
hectometer hm 100 m
decameter dam 10 m
meter m 1 m
decimeter dm 0.1 m
centimeter cm 0.01 m
millimeter mm 0.001 m

To convert an amount from one unit into another:

• If the original unit is less than the one we want to get, the amount will be divided by $10$ as many times as the number of rows that have to be “climbed” in the table above.
• If the original unit is larger than the one we want to get, the amount will be multiplied by $10$ as many times as the number of rows that have to be “gone down” in the table above.

If you want to convert $1400$ meters into decameters: One meter is less than a decameter therefore we have to divide $1400$ by $10$ once (because we have to go up once from the meter to decameter)

$\dfrac{1400}{10}=140$ decameters

That is, $1400$ meters are $140$ decameters.

## Measurements of mass

The unit for measuring mass is the gram. The other units that exist are:

Name Symbol Equivalence
kilogram kg 1000 g
hectogram hg 100 g
decagram dag 10 g
gram g 1 g
decigram dg 0.1 g
centigram cg 0.01 g
milligram mg 0.001 g

To convert an amount from one unit into another one:

• If the original unit is less than the one we want to get, the amount will be divided by $10$ as many times as the rows that have to be "climbed" in the table above.
• If the original unit is larger than the one we want to get, the amount will be multiplied by $10$ as many times as the rows that have to be "gone down" in the table above.

If we want to convert $23,4$ hectograms into decigrams:

An hectogram is greater than a decigram, therefore we have to multiply $23,4$ by $10$ three times given that in the above table we have to move down three rows to go from hectograms to decigrams.

Therefore:

$$23,4 \cdot 10 \cdot 10 \cdot 10 = 23.400$$ decigrams.

Namely, $23,4$ hectograms are $23.400$ decigrams.

## Measurements of capacity

To measure capacity the unit used is the liter. The following table shows other common measurements of capacity:

Name Symbol Equivalence
kiloliter kl 1000 l
hectoliter hl 100 l
decaliter dal 10 l
liter l 1 l
deciliter dl 0.1 l
centiliter cl 0.01 l
milliliter ml 0.001 l

To convert a number from one unit to another:

• If the original unit is less than the one we want to get, the amount will be divided by $10$ the same number of times as the number of rows that have to be “climbed” in the table above.
• If the original unit is larger than the one we want to get, the amount will be multiplied by $10$ the same number of times as the number of rows that have to be “gone down” in the table above.

If you want to convert $400$ milliliters to liters:

If we go from milliliters to liters we have to go up three rows, then we must divide by $10$ three times (which is the same as dividing by $1000$). Therefore:

$400:1000=0,4$ liters.

Namely $400$ milliliters are $0,4$ liters.

## Measurements of surface

To measure surfaces, the basic unit is the square meter, although the following units are also used:

Name Symbol Equivalence
square kilometer km2 1.000.000 m2
square hectometer hm2 10.000 m2
square decameter dam2 100 m2
square meter m2 1 m2
square decimeter dm2 0.01 m2
square centimeter cm2 0.0001 m2
square millimeter mm2 0.000001 m2

To switch a number from one unit to another:

• If the original unit is less than the one we want to get, the amount will be divided by $100$ the same number of times as the number of rows that have to be “climbed” in the table above.
• If the original unit is larger than the one we want to get, the amount will be multiplied by $100$ the same number of times as the number of rows that have to be “gone down” in the table above.

If you want to convert $0,003$ square kilometers to square decameters, then, in order to pass from square kilometers to square decametres, we move down two rows in the table above, therefore we must multiply by $100$ twice (or what is the same, per $10.000$) . Therefore:

$0,003\cdot10000=30$ square decameters.

Namely, $0,003$ square kilometers are $30$ square decameters.

## Measurements of volume

The most commonly used unit for measuring volume is the cubic meter. Other units commonly used are:

Name Symbol Equivalence
cubic kilometero km3 1.000.000.000 m3
cubic hectometer hm3 1.000.000 m3
cubic decameter dam3 1000 m3
cubic meter m3 1 m3
cubic decimeter dm3 0.001 m3
cubic centimeter cm3 0.000001 m3
cubic millimeter mm3 0.000000001 m3

To switch a number from one unit to another:

• If the original unit is less than the one we want to get, the amount will be divided by $1000$ the same number of times as the number of rows that have to be “climbed” in the table above.
• If the original unit is larger than the one we want to get, the amount will be multiplied by $1000$ the same number of times as the number of rows that have to be “gone down” in the table above.

If you want to convert $6.000.000$ cubic centimeters into cubic decimeters, you have to climb only one row, then it must divid it once by $1.000$:

$6.000.000:1.000=6.000$ cubic decimeters.

Therefore $6.000.000$ cubic centimeters are $6.000$ cubic decimeters.