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# Volume

How much space does a pen or an elephant take up? How much space do you take up? The volume of an object is something we might often refer to, but what is a volume exactly, how do we measure volumes, and what units do we use to describe a volume?Although the volume of something is a very intuitive notion,…

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# Volume

Volume
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Nie wieder prokastinieren mit unseren Lernerinnerungen. How much space does a pen or an elephant take up? How much space do you take up? The volume of an object is something we might often refer to, but what is a volume exactly, how do we measure volumes, and what units do we use to describe a volume?

## Definition of Volume

Although the volume of something is a very intuitive notion, it can be hard to describe exactly what a volume is. The following is a possible description of volume.

The volume of an object is a measure of the amount of 3-dimensional space it takes up.

This means that the volume of an elephant is larger than the volume of a mosquito.

A way of thinking about volume is asking how many sugar cubes would fit inside an object if it were hollow. If object $$1$$ would hypothetically contain $$200$$ sugar cubes and object $$2$$ would contain $$400$$, then object $$2$$ has a volume that is twice that of object $$1$$.

Another (non-countable but more precise) way of thinking about volume is how much water would fit inside an object if it were hollow. If you fill two objects with water and object $$1$$ is twice as heavy as object $$2$$, then object $$1$$ has twice as much volume as object $$2$$.

Just like mass, charge, and form, volume is a physical property of an object.

## Formula for Volume

There is no general formula for the volume of objects (if we don’t want to use calculus), but let’s look at a very basic object: a rectangular cuboid. This is the 3-dimensional version of a rectangle, see the figure below. A rectangular cuboid with sides a, b, and c, Arjan van Denzen - StudySmarter Originals.

It has sides of length $$a$$, $$b$$, and $$c$$. If we double $$a$$, then twice as many sugar cubes will fit inside the cuboid as before because we basically have two copies of the original cuboid on top of each other. This means that the volume of the cuboid doubles if we double the length $$a$$. The same goes for the lengths $$b$$ and $$c$$. These lengths are the only factors affecting the volume of the rectangular cuboid because they contain all the information necessary to define this object. So, the volume $$V_{\text{r.c.}}$$ of the rectangular cuboid must be a constant times the product of the length of all the sides, $$abc$$. It happens that the constant is $$1$$ so our formula becomes:

$V_{\text{r.c.}}=abc$

The volume of all other objects can now be defined via this cuboid: we make an object of which we want to know the volume. We make the object hollow and we fill it up with water. We then pour this water into a tank with a rectangular base such that the water takes the shape of a rectangular cuboid. We measure the three sides of the cuboid the water created and we multiply them to get the volume of our object.

The volume $$V_{\text{cube}}$$ of a cube with sides of length $$a$$ is the length of one side cubed, so $$V_{\text{cube}}=a^3$$ because a cube is just a rectangular cuboid with $$a=b=c$$.

## Measuring Volumes

We can also use water to actually measure the volume of objects in practice. We start with a completely full rectangular-cuboidal tank of water and dip our object in the water. Some of the water will overflow in this process because the water has to make room for the object to be inside the tank. This amount of room is the volume of the object. If we now remove the object from the water again, the water level in the tank will drop because we removed the volume of our object from the tank. The non-filled part of the tank now has the same volume as the object because we just took the object out of the tank! This non-filled part of the tank will have the form of a rectangular cuboid, so this volume is easy to measure, according to the formula we gave earlier. Voilà, this measured volume is the volume of our object. See the illustration below for a schematic presentation of this process. A way to measure the volume of objects, Arjan van Denzen - StudySmarter Originals.

## Dimensions of Volume in Physics

What are the dimensions of volume? Let’s take a look at the formula of the volume of our rectangular cuboid. We multiply three distances (from the 3 dimensions in the 3-dimensional space mentioned in the definition of volume) with each other to get a volume, so the dimensions of the volume of a rectangular cuboid must be $$\text{distance}^3$$. This automatically means that the dimensions of all volumes must be $$\text{distance}^3$$. The standard unit to measure a distance is the meter, so the standard unit to measure a volume is $$\mathrm{m}^3$$, or a cubic meter.

Another unit of volume that is often used is the liter. It has the symbol $$\mathrm{L}$$ and is defined as $$1\,\mathrm{L}=1\,\mathrm{dm}^3=10^{-3}\,\mathrm{m}^3$$.

A cube with sides of $$a=2$$ has a volume of $$8\,\mathrm{m}^3$$ because $$V=a^3=(2\,\mathrm{m})^3=8\,\mathrm{m}^3$$. This is $$8000\,\mathrm{L}$$.

## Calculation of Volumes

There are shapes for which the volume is reasonably easily calculated, i.e. without needing any advanced mathematics such as calculus every time you encounter such a shape.

Pyramids have a base and a height perpendicular to this base, see the figure below for an illustration. If the base of the pyramid has an area $$A$$ and the pyramid has a height $$h$$, then the volume $$V$$ of the pyramid is always given by $$V=Ah/3$$. A pyramid with height h and base area A, Arjan van Denzen - StudySmarter Originals.

The volume of a ball with radius $$r$$ is $$V=\dfrac{4}{3}\pi r^3$$.

Note how the dimensions of volume in both of the examples above work out to be $$\text{distance}^3$$.

If you ever calculate a volume and notice that it doesn’t have the right dimensions of $$\text{distance}^3$$, you have done something wrong. A volume always has dimensions of $$\text{distance}^3$$.

## Examples of Volumes in Physics

The volume of objects is important in a lot of physics questions.

Knowledge of the volume of a gas (for example, a gas held in a closed container) is essential for making conclusions about its density, pressure, and temperature. If we compress a gas to a smaller volume, its pressure will increase: it will push back on us.

Try squeezing a closed water bottle. You won’t get very far, because the decrease in volume of the air in the bottle will cause an increase in pressure, pushing back against you. This decrease in volume is essential for the force pushing back to increase.

When taking a bath, you have to take into account the volume of your body. Because your body takes the place of the water in the bathtub, the bathtub will overflow if your volume is larger than the volume of the non-filled part of the bathtub. Subconsciously, you take into account your own volume when filling up a bathtub.

## Volume - Key takeaways

• The volume of an object is a measure of the amount of 3-dimensional space it takes up.

• One way of thinking about volume is how much water would fit inside an object if it were hollow.

• The volume $$V$$ of a rectangular cuboid with sides $$a$$, $$b$$, and $$c$$ is given by $$V=abc$$.

• We can use a tank of water to measure the volume of objects.

• The standard unit of volume is the cubic meter ($$\mathrm{m}^3$$). A liter ($$\mathrm{L}$$) is $$\dfrac{1}{1000}$$ of a cubic meter.

• A volume always has dimensions of $$\text{distance}^3$$.

• The volume of a gas is often important when looking at gases in a physics context.

• The volume of your own body is important to take into account if you want to take a bath and you don’t want your bathtub to overflow.

In physics and other areas of science, the volume of an object is a measure of the amount of 3-dimensional space the object takes up.

The only general formula for the volume of an object is to integrate the volume form over the object, which can be regarded as the formal definition of a volume. Other than this higher-level formula, general simple formulas of volume do not exist.

In physics, the dimensions of volume are distance cubed. Therefore, the standard unit of volume is the cubic meter. Another popular unit of volume used in physics is the liter, which is a cubic decimeter.

Volume is a physical property of objects. However, materials do not have a fixed volume, as we can choose how much of such material we want to look at. You can ask how much volume a table has, but not how much volume wood has.

The volume of a cylinder is the area of one of its disks multiplied by its height. So a cylinder with height h and disk radius r has a volume of V=πr2h.

## Volume Quiz - Teste dein Wissen

Question

What does the volume of an object indicate?

How much 3-dimensional space the object takes up.

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Question

What is the volume of a cube with sides of $$30\text{ cm}$$?

$$V=(30\,\mathrm{cm})^3=(3\,\mathrm{dm})^3=27\,\mathrm{L}=0.027\,\mathrm{m}^3$$.

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Question

What is the volume $$V$$ of a rectangular cuboid with side lengths $$a$$, $$b$$, and $$c$$?

$$V=abc$$.

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Question

If we double two sides of a rectangular cuboid, what happens to its volume?

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Question

If we triple one side and halve the other two sides of a cube, what happens to its volume?

It is multiplied by 3/4.

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Question

How many liters is one cubic meter?

1000.

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Question

If we stretch a pyramid such that its base remains the same but its height triples, what happens to its volume?

It triples.

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Question

Why does the water level in the bathtub rise when you get in it?

You are adding your volume to the bathtub, so the water is displaced to the only place it can go: upwards.

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Question

If you squeeze a water bottle, what happens to its volume?

It decreases.

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Question

When blowing up a balloon, what happens to the volume of the air inside the balloon?

It increases.

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Question

What happens to the volume of your lungs when you exhale?

It decreases.

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Question

What happens to the volume of a cube if we triple the length of all sides?

It is multiplied by 27.

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Question

Is $$\mathrm{ft}^3$$ a possible unit of volume?

Yes.

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Question

Is $$\text{gallon}^3$$ a possible unit of volume?

No.

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