Measuring Volume
At Yaritza's corner store they bought a tank to store water: it says "2 m³". Her brother Wilkin thinks that is the same as 2 liters. Is he right? To find out, we need to know the units used to measure space well.
The Ladder of Volume Units
Volume is measured in cubic units: $1\ \text{cm}^3$ is the space taken up by a little cube with 1 cm edges. For length, each step is $\times 10$, but for volume each unit is 1,000 times the next one, because $10 \times 10 \times 10 = 1{,}000$.
Look at the conversion step diagram for units of volume in the figure. a) By what number do you multiply when going down a step? And by what number do you divide when…
Look at the conversion step diagram for units of volume in the figure.
a) By what number do you multiply when going down a step? And by what number do you divide when going up?
b) Using the diagram, express $2.4\ \text{m}^3$ in cubic decimeters and in cubic centimeters.
Convert to the specified units: a) $3\ \text{m}^3$ to cubic decimeters. b) $4\,500\ \text{dm}^3$ to cubic meters. c) $0.75\ \text{dm}^3$ to cubic centimeters.
Convert to the specified units:
a) $3\ \text{m}^3$ to cubic decimeters.
b) $4\,500\ \text{dm}^3$ to cubic meters.
c) $0.75\ \text{dm}^3$ to cubic centimeters.
Express in cubic decimeters: $3\ \text{m}^3\ 450\ \text{dm}^3\ 25\ \text{cm}^3$.
Express in cubic decimeters: $3\ \text{m}^3\ 450\ \text{dm}^3\ 25\ \text{cm}^3$.
In which unit would you express the volume of each of these objects? Choose between $\text{cm}^3$, $\text{dm}^3$, $\text{m}^3$ and $\text{hm}^3$, and justify your…
In which unit would you express the volume of each of these objects? Choose between $\text{cm}^3$, $\text{dm}^3$, $\text{m}^3$ and $\text{hm}^3$, and justify your choice:
a) A sugar cube.
b) A backpack.
c) The water in a reservoir.
d) A municipal swimming pool.
When converting $2.5\ \text{m}^3$ to cubic centimeters, Pablo writes in his notebook: $2.5\ \text{m}^3 = 2.5 \times 1\,000 = 2\,500\ \text{cm}^3$. a) Explain the mistake…
When converting $2.5\ \text{m}^3$ to cubic centimeters, Pablo writes in his notebook: $2.5\ \text{m}^3 = 2.5 \times 1\,000 = 2\,500\ \text{cm}^3$.
a) Explain the mistake he has made.
b) Carry out the correct conversion.
c) How many liters are $2.5\ \text{m}^3$?
Volume and Capacity
Capacity measures how much liquid fits inside something and is expressed in liters. Volume and capacity are, at heart, the same idea, and that is why they are related in such a simple way.
Now we can answer Wilkin: the $2\ \text{m}^3$ tank holds $2{,}000$ liters, a thousand times more than 2 liters!
Other units also appear in everyday life: one gallon is about $3.8$ liters. So a $2\ \text{m}^3$ tank is equivalent to about $528$ gallons.
Complete in your notebook: a) $75\ \text{l} =$ $\text{dm}^3$ b) $2\,500\ \text{cm}^3 =$ liters c) $0.5\ \text{m}^3 =$ liters
Complete in your notebook:
a) $75\ \text{l} =$ ____ $\text{dm}^3$
b) $2\,500\ \text{cm}^3 =$ ____ liters
c) $0.5\ \text{m}^3 =$ ____ liters
Out of the three proposed values for each volume, only one is sensible. Choose it and justify your answer: a) A soft drink can: $3.3\ \text{cm}^3$, $330\ \text{cm}^3$ or…
Out of the three proposed values for each volume, only one is sensible. Choose it and justify your answer:
a) A soft drink can: $3.3\ \text{cm}^3$, $330\ \text{cm}^3$ or $3.3\ \text{dm}^3$.
b) A washing machine: $55\ \text{dm}^3$, $250\ \text{dm}^3$ or $2.5\ \text{m}^3$.
c) A glass of water: $25$ ml, $250$ ml or $2.5$ l.
A water tank has a volume of $2.5\ \text{m}^3$. How many liters of water fit in it? How many $5$ liter containers could be filled?
A water tank has a volume of $2.5\ \text{m}^3$. How many liters of water fit in it? How many $5$ liter containers could be filled?
During a storm, $45$ liters per square meter of rain fell. a) How many millimeters of water depth is this equivalent to? b) How many liters fell on a garden of…
During a storm, $45$ liters per square meter of rain fell.
a) How many millimeters of water depth is this equivalent to?
b) How many liters fell on a garden of $60\ \text{m}^2$?
c) How many cubic meters is that?
A tank contains $1.2\ \text{m}^3$ of water. How many $750$ ml bottles can be filled with it?
A tank contains $1.2\ \text{m}^3$ of water. How many $750$ ml bottles can be filled with it?
An above-ground swimming pool is $3\ \text{m}$ long, $2\ \text{m}$ wide and $0.75\ \text{m}$ deep. a) How many liters of water fit in the pool? b) If it is filled using…
An above-ground swimming pool is $3\ \text{m}$ long, $2\ \text{m}$ wide and $0.75\ \text{m}$ deep.
a) How many liters of water fit in the pool?
b) If it is filled using a hose that pours $15$ liters per minute, how many hours does it take to fill up?
A garage with a surface area of $45\ \text{m}^2$ has flooded and the water has reached a height of $0.6\ \text{m}$. The water is pumped out with a pump that removes $45$…
A garage with a surface area of $45\ \text{m}^2$ has flooded and the water has reached a height of $0.6\ \text{m}$. The water is pumped out with a pump that removes $45$ liters per minute.
a) How many liters of water are in the garage?
b) How long will it take the pump to empty it? Express the result in hours.
State whether these statements are true or false, giving reasons: a) $1\ \text{dm}^3$ is equivalent to $1$ liter. b) $1\ \text{m}^3$ is equivalent to $100$ liters. c)…
State whether these statements are true or false, giving reasons:
a) $1\ \text{dm}^3$ is equivalent to $1$ liter.
b) $1\ \text{m}^3$ is equivalent to $100$ liters.
c) $1\ \text{cm}^3$ of water has a mass of $1$ gram.
d) To convert from $\text{m}^3$ to $\text{cm}^3$, multiply by $1\,000$.
Compound Form
Sometimes a volume is written split across several units, such as $3\ \text{m}^3\ 450\ \text{dm}^3$. That is the compound form: each part must be between 0 and 999 before moving on to the next unit.
Express in compound units: $12\,345\,678\ \text{mm}^3$.
Express in compound units: $12\,345\,678\ \text{mm}^3$.
Express in cubic meters and calculate: $2\ \text{dam}^3\ 15\ \text{m}^3 + 840\ \text{m}^3\ 250\ \text{dm}^3$. Express the result in hectoliters as well.
Express in cubic meters and calculate: $2\ \text{dam}^3\ 15\ \text{m}^3 + 840\ \text{m}^3\ 250\ \text{dm}^3$. Express the result in hectoliters as well.
An irrigation community consumes $1\ \text{dam}^3\ 260\ \text{m}^3$ of water in one week. a) Express this amount in liters. b) If the consumption is split equally over…
An irrigation community consumes $1\ \text{dam}^3\ 260\ \text{m}^3$ of water in one week.
a) Express this amount in liters.
b) If the consumption is split equally over the $7$ days of the week, how many liters are consumed per day?
c) Express the daily consumption in compound units.
A reservoir has a capacity of $0.45\ \text{km}^3$. After a year of drought, it is at $62\%$ of its capacity. a) How many cubic hectometers of water does it contain? b)…
A reservoir has a capacity of $0.45\ \text{km}^3$. After a year of drought, it is at $62\%$ of its capacity.
a) How many cubic hectometers of water does it contain?
b) How many liters is that? Express the result in scientific notation.
c) How many cubic hectometers are needed to fill it completely?
Volume, Capacity, and Mass
Water has a very useful property: 1 liter of water weighs 1 kg. And when an object is submerged in water, the volume it occupies is equal to the volume of water it displaces. This is how Archimedes discovered, more than 2,000 years ago, how to measure the volume of an irregular object without formulas.
Density relates mass and volume: $\text{mass} = \text{density} \times \text{volume}$. An object floats in water if its density is less than $1\ \text{kg/dm}^3$.
An empty tank weighs $12$ kg and, when full of diesel, weighs $522$ kg. If the density of diesel is $0.85\ \text{kg/dm}^3$, how many liters of diesel does the tank…
An empty tank weighs $12$ kg and, when full of diesel, weighs $522$ kg. If the density of diesel is $0.85\ \text{kg/dm}^3$, how many liters of diesel does the tank contain?
To measure the volume of a stone, we submerge it in a cylindrical glass containing water, as shown in the figure. The internal diameter of the glass is $8$ cm and the…
To measure the volume of a stone, we submerge it in a cylindrical glass containing water, as shown in the figure. The internal diameter of the glass is $8$ cm and the water level rises from $5$ cm to $8$ cm.
a) How much has the water level risen?
b) Calculate the volume of the stone in cubic centimeters (take $\pi \approx 3.14$).
A bathtub is full of water to the brim. Marta gets into it and $35$ liters of water overflow. a) What is the volume, in cubic decimeters, of the submerged part of…
A bathtub is full of water to the brim. Marta gets into it and $35$ liters of water overflow.
a) What is the volume, in cubic decimeters, of the submerged part of Marta's body?
b) If Marta weighs $32$ kg, what is the average density of her body in $\text{kg/dm}^3$? Compare it with the density of water ($1\ \text{kg/dm}^3$) and explain why she can float.
A block of cork has a volume of $25\ \text{dm}^3$ and a density of $0.24\ \text{kg/dm}^3$. a) What is its mass? b) Will it float in water? Give reasons for your answer…
A block of cork has a volume of $25\ \text{dm}^3$ and a density of $0.24\ \text{kg/dm}^3$.
a) What is its mass?
b) Will it float in water? Give reasons for your answer by comparing densities.
c) What volume of cork would be needed to have a mass of $30$ kg?
A concrete block of $1.5\ \text{m} \times 0.8\ \text{m} \times 0.4\ \text{m}$ has a density of $2\,400\ \text{kg/m}^3$. a) Calculate its volume and its mass. b) A crane…
A concrete block of $1.5\ \text{m} \times 0.8\ \text{m} \times 0.4\ \text{m}$ has a density of $2\,400\ \text{kg/m}^3$.
a) Calculate its volume and its mass.
b) A crane can lift up to $1.5$ metric tons. Can it move the block?
c) Could it move two blocks at the same time? Justify your answer.
Legend has it that Archimedes discovered how to check if the king's crown was made of pure gold by measuring the water it displaced when submerged. In the diagram, a…
Legend has it that Archimedes discovered how to check if the king's crown was made of pure gold by measuring the water it displaced when submerged. In the diagram, a crown weighing $750$ g is submerged in a measuring cylinder containing water.
a) What is the volume of the crown?
b) Calculate its density in $\text{g/cm}^3$.
c) Given that the density of gold is $19.3\ \text{g/cm}^3$ and that of silver is $10.5\ \text{g/cm}^3$, could the crown be solid gold? Give reasons for your answer.
A block of ice of $25\ \text{cm} \times 10\ \text{cm} \times 5\ \text{cm}$ has a density of $0.92\ \text{kg/dm}^3$. a) Calculate its volume in cubic decimeters. b) What…
A block of ice of $25\ \text{cm} \times 10\ \text{cm} \times 5\ \text{cm}$ has a density of $0.92\ \text{kg/dm}^3$.
a) Calculate its volume in cubic decimeters.
b) What is its mass?
c) When it melts, the resulting water has a density of $1\ \text{kg/dm}^3$. What volume will it occupy? Does it occupy more or less space than the ice? Why do you think this happens?
Back to the beginning: Wilkin was wrong: since $1\ \text{m}^3 = 1{,}000\ \text{dm}^3 = 1{,}000$ liters, a $2\ \text{m}^3$ tank holds $2{,}000$ liters.