---
title: "Measuring Length"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 7
exercises: 10
source: https://one-course.com/books/physics/1/en/chapter/7-measuring-length
---

# Chapter 7 — Measuring Length

Two doors, in two different houses: which one is taller? You cannot carry a door across town to stand it against the other. Last year we compared things side by side; this year we learn the greatest trick of science — turning a length into a *number* that can travel.

## 7.1 Comparing without numbers

**Example 7.1 (Side by side).**

To find the longer of two pencils, stand them together on the table: the one that sticks out wins. To compare your height with a friend’s, stand back to back. This works beautifully — as long as the two things can be brought together.

**Example 7.2 (The string trick).**

The two doors cannot meet — but a string can visit both. Stretch a string across the first door and cut it to exactly that height. Carry the string to the second door: if the string is too short, the second door is taller. The string carries the first door’s height across town. One step better: what if we could send the height in a *letter*? For that, we need numbers.

## 7.2 Measuring with steps and spans

**Definition 7.3 (Measuring a length).**

To *measure* a length is to count how many times a chosen *unit* — a step, a hand span, a small stick — fits along it, end to end with no gaps and no overlaps. The answer is a number of units: “the room is $12$ steps long.”

**Example 7.4 (Try it: pace the room).**

Cross your bedroom heel-to-toe, counting your foot-lengths. Ask a grown-up to do the same. You might count $18$ feet and the grown-up only $12$ — for the *same room*! Nobody counted wrong: your feet are shorter, so more of them fit. A measurement made with *my* foot cannot be checked with *your* foot.

![The same wall, paced twice: 12 small feet, or 8 big feet. Different feet, different numbers — the wall did not change.](https://one-course.com/images/onecourse/chapters/physics-1/g2-measuring-length/fig-542cce3ae460.svg)

*The same wall, paced twice: $12$ small feet, or $8$ big feet. Different feet, different numbers — the wall did not change.*

**Remark 7.5 (The argument in the marketplace).**

For a long time, people really did [measure](#def-g2-measuring-length-measure) in feet, thumbs and arm spans, and markets rang with arguments: whose foot? the tall merchant’s or the short customer’s? The fix, agreed upon all over the world, was to choose *one* [unit](#def-g2-measuring-length-measure) for everyone — a [unit](#def-g2-measuring-length-measure) that belongs to nobody’s body.

## 7.3 The centimetre and the metre

**Definition 7.6 (Centimetre and metre).**

The *centimetre* (written $\mathrm{cm}$) is a small [unit](#def-g2-measuring-length-measure) of length, about the width of a fingernail; it is printed on every ruler. The *metre* (written $\mathrm{m}$) is a big [unit](#def-g2-measuring-length-measure), exactly $100$ centimetres, the length of a very big step. These [units](#def-g2-measuring-length-measure) are the same for everyone on Earth: your $20\,\mathrm{cm}$ and my $20\,\mathrm{cm}$ are exactly equal.

![A metre stick: one hundred centimetres in a row. The marks let us count them quickly instead of one by one.](https://one-course.com/images/onecourse/chapters/physics-1/g2-measuring-length/fig-2ea82f49608c.svg)

*A [metre](#def-g2-measuring-length-units) stick: one hundred [centimetres](#def-g2-measuring-length-units) in a row. The marks let us count them quickly instead of one by one.*

**Method 7.7 (Measuring with a ruler).**

1. Lay the ruler along the object, touching it;
2. put the ruler’s *zero mark* exactly at one end of the object — not the ruler’s edge, the $0$ ;
3. read the number at the other end: that is the length in [centimetres](#def-g2-measuring-length-units) .

If the end lands between two marks, say “between $7\,\mathrm{cm}$ and $8\,\mathrm{cm}$, closer to $8$” — honest words beat a made-up number.

![Reading a ruler: zero mark at one end of the crayon, read the other end. This crayon is 8\, cm long.](https://one-course.com/images/onecourse/chapters/physics-1/g2-measuring-length/fig-3a8dcaa3d7c8.svg)

*Reading a ruler: zero mark at one end of the crayon, read the other end. This crayon is $8\,\mathrm{cm}$ long.*

![Measuring done properly: the pencil’s end sits exactly on the ruler’s zero mark.](https://one-course.com/images/onecourse/chapters/physics-1/g2-measuring-length/img-8dda5027bc28.jpg)

*Measuring done properly: the pencil’s end sits exactly on the ruler’s zero mark.*

**Example 7.8 (Which unit for which job?).**

Small things like a beetle, a stamp or your hand are measured in [centimetres](#def-g2-measuring-length-units). Big things like a corridor, a bus or a swimming pool are measured in [metres](#def-g2-measuring-length-units): the pool is $25\,\mathrm{m}$ long — imagine counting that in fingernail-widths! Choosing a sensible [unit](#def-g2-measuring-length-measure) is half the work of measuring.

**Example 7.9 (Estimate, then check).**

Before measuring, players guess: how long is the kitchen table? Papa says $2\,\mathrm{m}$, Lina says $1\,\mathrm{m}$ and a bit. Then the [metre](#def-g2-measuring-length-units) stick decides: $1\,\mathrm{m}$ and almost a half — point for Lina. Guessing first trains your eye; measuring keeps everyone honest.

## 7.4 Exercises

**Exercise 7.1 ★.**

How can you compare your height with a friend’s without any ruler? And with the height of a friend who lives in another town?

**Solution of Exercise 7.1.**

Nearby: stand back to back. Far away: [measure](#def-g2-measuring-length-measure) yourself in [centimetres](#def-g2-measuring-length-units) and send the number — a number can travel in a letter, a back-to-back comparison cannot.

**Exercise 7.2 ★.**

Mara [measures](#def-g2-measuring-length-measure) the classroom in steps and finds $20$; her teacher finds $14$. Who counted wrong? Explain.

**Solution of Exercise 7.2.**

Neither counted wrong. Mara’s steps are shorter, so more of them fit in the same classroom. Different [units](#def-g2-measuring-length-measure) give different numbers for the same length.

**Exercise 7.3 ★.**

What is special about the [centimetre](#def-g2-measuring-length-units), compared to a foot-length or a hand span? Why did people agree on it?

**Solution of Exercise 7.3.**

The [centimetre](#def-g2-measuring-length-units) is the same for everyone — it belongs to nobody’s body. A measurement in [centimetres](#def-g2-measuring-length-units) can be checked by any other person with any ruler.

**Exercise 7.4 ★.**

Which [unit](#def-g2-measuring-length-measure) fits better, $\mathrm{cm}$ or $\mathrm{m}$: the length of an ant; the height of a door; the length of the schoolyard; the width of this book?

**Solution of Exercise 7.4.**

The ant: $\mathrm{cm}$ (even less!); the door: $\mathrm{m}$; the schoolyard: $\mathrm{m}$; the width of this book: $\mathrm{cm}$.

**Exercise 7.5 ★.**

Tim puts the edge of his ruler — not the zero mark — against the end of his pencil, and reads $13\,\mathrm{cm}$. Is his pencil really $13\,\mathrm{cm}$ long? What did he forget from [Method 7.7](#met-g2-measuring-length-ruler)?

**Solution of Exercise 7.5.**

No — there is usually a small blank space between a ruler’s edge and its zero mark, so the pencil is a little shorter than $13$ [centimetres](#def-g2-measuring-length-units). Tim forgot to put the *zero mark*, not the edge, at the pencil’s end.

**Exercise 7.6 ★.**

Measure with a ruler: the width of your hand; the length of a spoon; the height of a mug. Write each answer with its [unit](#def-g2-measuring-length-measure).

**Solution of Exercise 7.6.**

Answers vary; typical: hand about $7\,\mathrm{cm}$ wide, spoon about $15\,\mathrm{cm}$, mug about $9\,\mathrm{cm}$. The important part: each number carries its $\mathrm{cm}$.

**Exercise 7.7 ★.**

A door is $2\,\mathrm{m}$ tall. How many [centimetres](#def-g2-measuring-length-units) is that? (Remember: $1\,\mathrm{m}$ is $100$ $\mathrm{cm}$.)

**Solution of Exercise 7.7.**

$100 + 100 = 200$: the door is $200\,\mathrm{cm}$ tall.

**Exercise 7.8 ★.**

First guess, then [measure](#def-g2-measuring-length-measure): how many [metres](#def-g2-measuring-length-units) long is your bed? Was your guess too big, too small, or nearly right?

**Solution of Exercise 7.8.**

Answers vary; most beds are close to $2\,\mathrm{m}$. The point is to compare the guess with the measured number.

**Exercise 7.9 ★★.**

A ribbon is $45\,\mathrm{cm}$ long, another is $38\,\mathrm{cm}$ long. Which is longer, and by how many [centimetres](#def-g2-measuring-length-units)? Could you have decided *without* numbers? Which way is easier to write in a letter?

**Solution of Exercise 7.9.**

The $45\,\mathrm{cm}$ ribbon is longer: $45 - 38 = 7$, so by $7\,\mathrm{cm}$. Without numbers, laying them side by side also finds the longer one — but “$45\,\mathrm{cm}$” fits in a letter, and “side by side” does not.

**Exercise 7.10 ★★.**

Grandpa says: “My garden is $30$ paces long.” Explain to Grandpa, kindly, why “$30$ paces” cannot be checked by someone else, and what he should use instead. What would happen if a small child paced the same garden?

**Solution of Exercise 7.10.**

Nobody else has Grandpa’s legs: a small child pacing the same garden might count $60$ little paces, and a visitor could not check “$30$ paces” without Grandpa there. Measured in [metres](#def-g2-measuring-length-units) — the same for everyone — the garden’s length can be written down, sent, and checked by anybody.