---
title: "Mass and Volume"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 38
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/38-mass-and-volume
---

# Chapter 38 — Mass and Volume

How much lemonade fits in the bottle, and how heavy is the crate of bottles? Two different questions — room and stuff — that everyday talk cheerfully tangles into one word, “big”. Science untangles them: [volume](#def-g6-mass-and-volume-volume) for the room a thing takes, [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) for the stuff it holds. Measuring the second, you are an old hand; today we learn to [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) the first — even for a pebble with no shape a ruler could love.

## 38.1 Volume: the room a thing takes

**Definition 38.1 (Volume).**

The *volume* of a thing is the amount of room it takes up. [Solids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-solid), [liquids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) and gases all have volume: the brick takes its room, the lemonade takes the bottle’s inside, the [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) takes the rest. Volume is measured in *cubic centimetres* ($\mathrm{cm}^{3}$) — the room of a little cube one [centimetre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) along each edge — and in *litres* ($\mathrm{L}$).

**Proposition 38.2 (The litre in cubes).**

The two families of [volume](#def-g6-mass-and-volume-volume) [units](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) are one family:

$$
1\,\mathrm{L} = 1000\,\mathrm{cm}^{3},
\qquad
1\,\mathrm{mL} = 1\,\mathrm{cm}^{3}.
$$

A [litre](#def-g6-mass-and-volume-volume) is exactly the room inside a cube ten [centimetres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) along each edge: $10 \times 10 \times 10 = 1000$ little cubes.

![One litre, unpacked: a thousand centimetre-cubes stacked ten by ten by ten.](https://one-course.com/images/onecourse/chapters/physics-1/g6-mass-and-volume/fig-69a6a9d7bf41.svg)

*One [litre](#def-g6-mass-and-volume-volume), unpacked: a thousand centimetre-cubes stacked ten by ten by ten.*

**Example 38.3 (Volumes to know by heart).**

A teaspoon holds about $5\,\mathrm{mL}$; a drinking glass about $25\,\mathrm{cL}$, a quarter [litre](#def-g6-mass-and-volume-volume); the milk carton, $1\,\mathrm{L}$; a bathtub, around $150\,\mathrm{L}$; a sugar cube takes about $3\,\mathrm{cm}^{3}$; a die about $4\,\mathrm{cm}^{3}$. Anchors like these make the judging step of every measurement possible.

## 38.2 Measuring liquids

**Method 38.4 (Reading a measuring cylinder).**

The laboratory’s jug is the *measuring cylinder*: tall, narrow, finely graduated.

1. set the cylinder on a flat table — never read it in your hand;
2. find the graduation value, as with every scale;
3. bring your eye level with the [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) ’s surface: the surface is not flat but slightly curved — a *meniscus* — climbing a little up the glass walls;
4. read at the *bottom* of the curve, straight across.

Eye too high or too low, and the curve lies to you by a step or two — the slanted-view mistake in its favorite disguise.

![The meniscus: liquids climb a little up the glass. The honest reading is at the bottom of the curve, eye level.](https://one-course.com/images/onecourse/chapters/physics-1/g6-mass-and-volume/fig-008f89062dae.svg)

*The [meniscus](#met-g6-mass-and-volume-cylinder): [liquids](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) climb a little up the glass. The honest reading is at the bottom of the curve, eye level.*

![The meniscus up close: read the flat middle of the curve, with your eye at its level.](https://one-course.com/images/onecourse/chapters/physics-1/g6-mass-and-volume/img-7b7e23bc8c45.jpg)

*The [meniscus](#met-g6-mass-and-volume-cylinder) up close: read the flat middle of the curve, with your eye at its level.*

## 38.3 Measuring solids — even lumpy ones

A brick-shaped [solid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-solid) surrenders to the ruler: length times width times height gives its [volume](#def-g6-mass-and-volume-volume) in $\mathrm{cm}^{3}$, as your mathematics course showed. But what of a pebble, a key, a plum? No ruler fits a lump. The answer is two thousand years old, and it came out of a bathtub.

**Proposition 38.5 (Displacement).**

A [solid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-solid) lowered completely underwater pushes aside — displaces — exactly its own [volume](#def-g6-mass-and-volume-volume) of water. The water level’s rise [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) the intruder: lumpy or smooth, the water molds itself to every dent and bump.

**Method 38.6 (Volume by the rise of water).**

1. pour water into a measuring cylinder — enough to cover the object — and read the [volume](#def-g6-mass-and-volume-volume) : say $120\,\mathrm{mL}$ ;
2. tie a thread to the object and lower it gently until fully underwater, no splash, no touching the walls;
3. read again: say $145\,\mathrm{mL}$ ;
4. subtract: $145 - 120 = 25$ — the object’s [volume](#def-g6-mass-and-volume-volume) is $25\,\mathrm{cm}^{3}$ (millilitres of rise, centimetre-cubes of stone: the same thing).

![The rise method: the water climbs by exactly the pebble’s volume — 145 - 120 = 25\, cm3 of pebble.](https://one-course.com/images/onecourse/chapters/physics-1/g6-mass-and-volume/fig-6c8e53c8c640.svg)

*The rise method: the water climbs by exactly the pebble’s [volume](#def-g6-mass-and-volume-volume) — $145 - 120 = 25\,\mathrm{cm}^{3}$ of pebble.*

**Example 38.7 (The crown in the bathtub).**

The old story: a king suspected his goldsmith had thinned the royal crown’s gold with silver, and asked the scientist Archimedes to find out without harming the crown. Stepping into a full bath, Archimedes watched the water slosh over the rim — and saw in a flash that the overflow measured his own body’s [volume](#def-g6-mass-and-volume-volume), lumps and all. Legend says he ran through the streets shouting “Eureka!” — *I have found it!* How the crown’s [volume](#def-g6-mass-and-volume-volume) unmasked the fraud needs one more idea, two chapters ahead; the bathtub gave him the [volume](#def-g6-mass-and-volume-volume) of *anything*.

## 38.4 Mass and volume are different answers

**Example 38.8 (Same volume, different mass).**

Fill two identical $25\,\mathrm{cL}$ glasses, one with water, one with honey, and weigh them (subtracting the glasses): the honey glass holds noticeably more [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) in the very same room. Same [volume](#def-g6-mass-and-volume-volume), different [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass). And a [litre](#def-g6-mass-and-volume-volume) of [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) weighs hardly more than a [gram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units), in the room where a [litre](#def-g6-mass-and-volume-volume) of water weighs a full [kilogram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units). Room and stuff are truly different bookkeepers — hold that thought firmly: it becomes a law, with a name, in the last chapter of this year.

**Remark 38.9 (Weighing what cannot stand alone).**

How to weigh flour without weighing its bowl? Modern balances have a button for it: set the empty bowl on the pan, press — the display resets to zero — then pour: the balance now reports the flour alone. The maneuver is called *taring*. Without the button, weigh twice and subtract: bowl-with-flour minus empty bowl. Either way, it is the fair-start rule again: begin from an honest zero.

## 38.5 Exercises

**Exercise 38.1 ★.**

What question does [volume](#def-g6-mass-and-volume-volume) answer, and what question does [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) answer? Give the two chief [units](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) of each.

**Solution of Exercise 38.1.**

[Volume](#def-g6-mass-and-volume-volume): how much room a thing takes — $\mathrm{cm}^{3}$ and [litres](#def-g6-mass-and-volume-volume). [Mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass): how much stuff it holds — [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) and [kilograms](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units).

**Exercise 38.2 ★.**

Convert: $2\,\mathrm{L}$ to $\mathrm{cm}^{3}$; $350\,\mathrm{mL}$ to $\mathrm{cm}^{3}$; $1500\,\mathrm{cm}^{3}$ to [litres](#def-g6-mass-and-volume-volume); half a [litre](#def-g6-mass-and-volume-volume) to millilitres.

**Solution of Exercise 38.2.**

$2000\,\mathrm{cm}^{3}$; $350\,\mathrm{cm}^{3}$; $1.5\,\mathrm{L}$; $500\,\mathrm{mL}$.

**Exercise 38.3 ★.**

Why must a measuring cylinder be read at eye level, on the table, at the bottom of the [meniscus](#met-g6-mass-and-volume-cylinder)? Name the mistake each rule prevents.

**Solution of Exercise 38.3.**

On the table: a tilted cylinder tilts the surface. Eye level: a high or low eye shifts the apparent mark (the slanted-view mistake). Bottom of the [meniscus](#met-g6-mass-and-volume-cylinder): the [liquid](https://one-course.com/books/physics/1/en/chapter/14-solids-liquids-gases#def-g3-solids-liquids-gases-liquid) climbs the walls, and only the curve’s bottom stands at the true [volume](#def-g6-mass-and-volume-volume).

**Exercise 38.4 ★.**

A cylinder reads $200\,\mathrm{mL}$; with a plum lowered in, it reads $265\,\mathrm{mL}$. What is the plum’s [volume](#def-g6-mass-and-volume-volume), in $\mathrm{cm}^{3}$?

**Solution of Exercise 38.4.**

$265 - 200 = 65$: the plum’s [volume](#def-g6-mass-and-volume-volume) is $65\,\mathrm{cm}^{3}$.

**Exercise 38.5 ★.**

Why does the rise method work even for a lumpy pebble, when no ruler could [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) it? Which proposition answers?

**Solution of Exercise 38.5.**

The water molds itself to every dent and bump, so the pushed-aside water equals the pebble’s [volume](#def-g6-mass-and-volume-volume) exactly — the displacement proposition. The ruler needs straight edges; the water needs none.

**Exercise 38.6 ★.**

A brick-shaped eraser [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) $5\,\mathrm{cm}$ by $2\,\mathrm{cm}$ by $1\,\mathrm{cm}$. Its [volume](#def-g6-mass-and-volume-volume)? Check-plan: how would you confirm it with the rise method?

**Solution of Exercise 38.6.**

$5 \times 2 \times 1 = 10\,\mathrm{cm}^{3}$. Check: lower it into a cylinder — the reading should rise by exactly $10\,\mathrm{mL}$.

**Exercise 38.7 ★.**

To weigh rice: empty bowl $280\,\mathrm{g}$; bowl with rice $745\,\mathrm{g}$. How much rice? What button does the same job in one step?

**Solution of Exercise 38.7.**

$745 - 280 = 465\,\mathrm{g}$ of rice. The tare button: zero the balance with the empty bowl on it, then pour.

**Exercise 38.8 ★.**

Which has the greater [volume](#def-g6-mass-and-volume-volume), a [kilogram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) of feathers or a [kilogram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) of iron? Which has the greater [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass)? (Answer both carefully.)

**Solution of Exercise 38.8.**

Equal masses — one [kilogram](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) each. The feathers take far more room: much greater [volume](#def-g6-mass-and-volume-volume) for the same stuff.

**Exercise 38.9 ★★.**

The rise method fails for a cork: it [floats](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words). Propose a fix that still uses the cylinder — and say what must be subtracted if your fix uses a helper object.

**Solution of Exercise 38.9.**

[Sink](https://one-course.com/books/physics/1/en/chapter/5-floating-and-sinking#def-g1-floating-and-sinking-words) the cork on purpose: push it under with a thin skewer (read the rise while only the cork is submerged), or tie it to a heavy sinker — [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) the sinker’s rise alone first, then the pair together, and subtract the sinker’s share.

**Exercise 38.10 ★★.**

A juice carton claims $1\,\mathrm{L}$. Poured out, it fills a $250\,\mathrm{mL}$ glass three times, and the fourth glass only up to the $200\,\mathrm{mL}$ mark. Was the claim honest? By how much?

**Solution of Exercise 38.10.**

Poured out: $3 \times 250 + 200 = 950\,\mathrm{mL}$. The claim missed by $50\,\mathrm{mL}$ — a twentieth of a [litre](#def-g6-mass-and-volume-volume) short.

**Exercise 38.11 ★★.**

A necklace is lowered into a cylinder reading $80.0\,\mathrm{mL}$; it rises to $83.5\,\mathrm{mL}$. The jeweler claims the necklace contains $5\,\mathrm{cm}^{3}$ of gold. Could the claim be true? What, exactly, did the water [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring)?

**Solution of Exercise 38.11.**

The rise is $83.5 - 80.0 = 3.5\,\mathrm{mL}$: the necklace’s whole [volume](#def-g6-mass-and-volume-volume) is $3.5\,\mathrm{cm}^{3}$. The water [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) total [volume](#def-g6-mass-and-volume-volume), metal and clasp and all — so a claim of $5\,\mathrm{cm}^{3}$ of gold cannot be true: there is not even $5\,\mathrm{cm}^{3}$ of necklace.

**Exercise 38.12 ★★★.**

Design a bathtub-style check of [Proposition 38.5](#prop-g6-mass-and-volume-displacement) itself, using a brick-shaped object whose [volume](#def-g6-mass-and-volume-volume) the ruler already knows. Describe the steps, predict the two cylinder readings for a $4 \times 3 \times 2$ [centimetre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) block lowered into $150\,\mathrm{mL}$, and state what result would prove the proposition wrong.

**Solution of Exercise 38.12.**

The ruler promises $4 \times 3 \times 2 = 24\,\mathrm{cm}^{3}$. Lower the block (thread, no splash, fully under, not touching the walls) into $150\,\mathrm{mL}$: the prediction is a rise from $150\,\mathrm{mL}$ to $174\,\mathrm{mL}$. If the water rose by anything other than $24\,\mathrm{mL}$ — outside the cylinder’s honest step or two — the displacement proposition would stand refuted. (It never has been.)

## 38.6 Problem: Bottling Day at the Orchard

**Problem 38.1.**

Weekend problem — pressing day at the orchard; litres into bottles, crates onto scales; the pebble in the jug

The orchard presses its apples today, and every helper gets a measuring job.

**Part I — Juice by the [litre](#def-g6-mass-and-volume-volume).** The press has filled a $20\,\mathrm{L}$ barrel.

1. How many [cubic centimetres](#def-g6-mass-and-volume-volume) is $20\,\mathrm{L}$ ?
2. The juice is bottled in $75\,\mathrm{cL}$ bottles. How many *full* bottles does the barrel give?
3. How much juice is left over after the last full bottle, in centilitres?
4. The leftover is poured into glasses of $25\,\mathrm{cL}$ . How many glasses does it fill?

**Part II — Crates on the scale.**

5. An empty bottle has [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) $450\,\mathrm{g}$ . Using Part I, what is the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of the juice in one full bottle, if a [litre](#def-g6-mass-and-volume-volume) of juice has a [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of about $1\,\mathrm{kg}$ ? (Work in [grams](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-units) .)
6. So what does one full bottle weigh altogether?
7. A wooden crate weighs $2\,\mathrm{kg}$ empty and carries six full bottles. Total [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of a loaded crate?
8. The van may carry $200\,\mathrm{kg}$ of crates. How many loaded crates may it take? (Whole crates only.)

**Part III — The pebble in the jug.** Little Tom drops a pebble into a full $1\,\mathrm{L}$ jug of juice, and juice overflows onto the table.

9. The mopped-up overflow [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) $40\,\mathrm{mL}$ . What did the spill [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) about the pebble, and by which proposition?
10. After the pebble is fished out, how much juice remains in the jug?
11. Tom protests: “The pebble is small — look, it hides in my fist!” His sister answers with a measuring cylinder: she puts the pebble in $100\,\mathrm{mL}$ of water. What reading proves the spill told the truth?
12. Bonus bookkeeping: the pebble weighs $104\,\mathrm{g}$ on the kitchen scale. Without any new idea, can you yet say whether pebble-stuff is heavier or lighter than juice-stuff *for the same room taken* ? Compare $104\,\mathrm{g}$ of pebble in $40\,\mathrm{cm}^{3}$ with what [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) of juice would fill those same $40\,\mathrm{cm}^{3}$ — and keep your conclusion warm for the density chapter.

**Solution of Problem 38.1.**

**1.** $20 \times 1000 = 20\,000\,\mathrm{cm}^{3}$. **2.** $20\,\mathrm{L} = 2000\,\mathrm{cL}$; $2000 \div 75 = 26$ full bottles (with remainder). **3.** $26 \times 75 = 1950\,\mathrm{cL}$, so $2000 - 1950 =
50\,\mathrm{cL}$ remain. **4.** $50 \div 25 = 2$ glasses, exactly. **5.** $75\,\mathrm{cL} = 0.75\,\mathrm{L}$, and a [litre](#def-g6-mass-and-volume-volume) weighs about $1000\,\mathrm{g}$: the juice is about $750\,\mathrm{g}$. **6.** $750 + 450 = 1200\,\mathrm{g}$ — $1.2\,\mathrm{kg}$. **7.** $6 \times 1.2 = 7.2\,\mathrm{kg}$ of bottles plus $2\,\mathrm{kg}$ of crate: $9.2\,\mathrm{kg}$. **8.** $200 \div 9.2$: $21$ crates would be $193.2\,\mathrm{kg}$, $22$ would be $202.4\,\mathrm{kg}$ — too much. The van takes $21$ crates. **9.** The spill is the juice the pebble displaced: the pebble’s [volume](#def-g6-mass-and-volume-volume) is $40\,\mathrm{cm}^{3}$, by the displacement proposition. **10.** $1000 - 40 = 960\,\mathrm{mL}$ of juice remain. **11.** The cylinder should climb from $100\,\mathrm{mL}$ to $140\,\mathrm{mL}$ — a $40\,\mathrm{mL}$ rise, matching the spill. Fists [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) nothing; water tells the truth. **12.** Those $40\,\mathrm{cm}^{3}$, filled with juice, would hold about $40\,\mathrm{g}$ of juice — the pebble packs $104\,\mathrm{g}$ into the same room: pebble-stuff is heavier than juice-stuff for equal room, more than twice over. The density chapter will give this comparison its proper name and number.