---
title: "Measures and Perimeter"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 28
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter
---

# Chapter 28 — Measures and Perimeter

Grade 3 measured with rulers, scales and clocks ([Chapter 19](https://one-course.com/books/math/1/en/chapter/19-measuring#ch-g3-measure)); this chapter adds the units table — the machine that converts everything — and two ways of measuring a flat shape: the length of its border ([perimeter](#def-g4-measure-perimeter)) and the room inside it ([area](#def-g4-measure-area), counted in squares for now).

## 28.1 The units table

**Method 28.1 (Converting with the table).**

For lengths, each column is worth ten of the next:

| km | hm | dam | m | dm | cm | mm |
| --- | --- | --- | --- | --- | --- | --- |
|  |  |  | $3$ | $4$ | $0$ |  |

1. Write the number with one digit per column, the units digit of the measure in its unit’s column (here $3.4$ m: the $3$ under m);
2. read the number in the new unit: $3.4$ m $= 340$ cm (fill empty columns with [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) );
3. masses (kg … g) and capacities (L … cL) work the same way.

**Example 28.2.**

$5$ km $= 5\,000$ m; $270$ cm $= 2.7$ m (the $2$ lands under m); $3$ kg $250$ g $= 3\,250$ g; $2.5$ L $= 250$ cL.

## 28.2 Perimeter

**Definition 28.3 (Perimeter).**

The *perimeter* of a [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) is the total length of its border: add the lengths of all its sides (in the same unit!). Shortcuts for the classics:

$$
\text{rectangle: } P = 2 \times (L + w),
\qquad
\text{square: } P = 4 \times c .
$$

**Example 28.4.**

A rectangular garden $12$ m by $7$ m:

$$
P = 2 \times (12 + 7) = 2 \times 19 = 38 \text{ m}
$$

of fence. A square photo frame of side $18$ cm: $P = 4 \times 18 = 72$ cm.

## 28.3 Area: counting squares

**Definition 28.5 (Area by counting).**

The *area* of a figure drawn on a grid is the number of grid squares it covers. Two half-squares count as one. Area answers “how much surface”, [perimeter](#def-g4-measure-perimeter) answers “how long is the border” — two different questions!

![Two figures on the grid: areas 8 and 6 squares. Now walk the borders and count the units: 12 for both! Same perimeter, different areas.](https://one-course.com/images/onecourse/chapters/math-1/g4-measure/fig-d734f864697f.svg)

*Two figures on the grid: [areas](#def-g4-measure-area) $8$ and $6$ squares. Now walk the borders and count the units: $12$ for both! Same [perimeter](#def-g4-measure-perimeter), different [areas](#def-g4-measure-area).*

**Example 28.6 (Half-squares).**

A right triangle drawn on the grid with legs $4$ and $2$ covers $3$ full squares and $2$ half-squares … count carefully: its [area](#def-g4-measure-area) is [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the $4 \times 2$ rectangle, i.e. $4$ squares. Cutting a rectangle along its diagonal always gives two triangles of equal [area](#def-g4-measure-area).

**Example 28.7 (Same area, different perimeters).**

A $4 \times 4$ square and an $8 \times 2$ rectangle both cover $16$ squares — same [area](#def-g4-measure-area). Borders: $16$ units for the square, $20$ for the rectangle. Neither measure determines the other: a long thin shape has lots of border for little surface.

## 28.4 Durations

**Method 28.8 (Adding times across the hour).**

To add or subtract durations, work in hops through the full hours (recall $1$ h $= 60$ min):

$$
9\text{ h }40 + 35 \text{ min}:
\quad 9\text{ h }40 \xrightarrow{+20} 10\text{ h }00
\xrightarrow{+15} 10\text{ h }15 .
$$

Never add minutes past $60$ without converting: $40 + 35 = 75$ min $= 1$ h $15$ min.

**Example 28.9.**

The train leaves at $8{:}47$ and arrives at $11{:}05$. Duration: $8{:}47 \to 9{:}00$ is $13$ min; $9{:}00 \to 11{:}00$ is $2$ h; $11{:}00 \to 11{:}05$ is $5$ min. Total: $2$ h $18$ min.

## 28.5 Exercises

**Exercise 28.1 ★.**

Convert with the units table: $7$ m in cm; $3.2$ km in m; $85$ mm in cm; $4\,500$ m in km.

**Solution of Exercise 28.1.**

$7$ m $= 700$ cm; $3.2$ km $= 3\,200$ m; $85$ mm $= 8.5$ cm; $4\,500$ m $= 4.5$ km.

**Exercise 28.2 ★.**

Convert: $2$ kg in g; $1\,800$ g in kg and g; $3.5$ L in cL; $25$ cL in L.

**Solution of Exercise 28.2.**

$2$ kg $= 2\,000$ g; $1\,800$ g $= 1$ kg $800$ g; $3.5$ L $= 350$ cL; $25$ cL $= 0.25$ L.

**Exercise 28.3 ★.**

Compute the [perimeter](#def-g4-measure-perimeter): a rectangle $9$ cm $\times$ $5$ cm; a square of side $7.5$ cm; a triangle with sides $6$ cm, $8$ cm and $10$ cm.

**Solution of Exercise 28.3.**

Rectangle: $2 \times (9 + 5) = 28$ cm. Square: $4 \times 7.5 = 30$ cm. Triangle: $6 + 8 + 10 = 24$ cm.

**Exercise 28.4 ★.**

A rectangle has [perimeter](#def-g4-measure-perimeter) $26$ cm and length $8$ cm. Find its width (first find length $+$ width).

**Solution of Exercise 28.4.**

Length $+$ width $= 26 \div 2 = 13$ cm, so the width is $13 - 8 = 5$ cm.

**Exercise 28.5 ★.**

On grid paper, draw an L-shaped figure covering exactly $10$ squares, and count the units of its [perimeter](#def-g4-measure-perimeter).

**Solution of Exercise 28.5.**

Many L-shapes work; for a $10$-square L made of a $4 \times 2$ rectangle plus a $2 \times 1$ foot, walking the border gives $16$ units. (Any correct figure and careful count is right; the [perimeter](#def-g4-measure-perimeter) depends on the L chosen.)

**Exercise 28.6 ★.**

Draw two different rectangles of [area](#def-g4-measure-area) $12$ squares. Compute the [perimeter](#def-g4-measure-perimeter) of each. Which is “rounder”, which is “thinner”?

**Solution of Exercise 28.6.**

$3 \times 4$: [perimeter](#def-g4-measure-perimeter) $14$. $2 \times 6$: [perimeter](#def-g4-measure-perimeter) $16$. ($1 \times 12$: [perimeter](#def-g4-measure-perimeter) $26$.) The closer to a square, the smaller the [perimeter](#def-g4-measure-perimeter) for the same [area](#def-g4-measure-area).

**Exercise 28.7 ★.**

A right triangle is drawn on the grid with legs $6$ and $3$. What is its [area](#def-g4-measure-area) in squares? (Use [Example 28.6](#ex-g4-measure-half).)

**Solution of Exercise 28.7.**

[Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the $6 \times 3$ rectangle: $18 \div 2 = 9$ squares.

**Exercise 28.8 ★.**

Compute the durations: from $10{:}20$ to $12{:}45$; from $7{:}35$ to $9{:}10$. A film lasts $1$ h $50$ and starts at $20{:}30$: when does it end?

**Solution of Exercise 28.8.**

$10{:}20 \to 12{:}45$: $2$ h $25$ min. $7{:}35 \to 9{:}10$: $1$ h $35$ min. Film: $20{:}30 + 1$ h $50 = 22{:}20$.

**Exercise 28.9 ★.**

Kim runs $3$ laps of a rectangular field $60$ m by $45$ m. How many meters does she run?

**Solution of Exercise 28.9.**

One lap: $2 \times (60 + 45) = 210$ m. Three laps: $3 \times 210 = 630$ m.

**Exercise 28.10 ★★.**

A farmer wants to fence a square field of side $85$ m, leaving a gate of $4$ m without fence. How many meters of fence must he buy?

**Solution of Exercise 28.10.**

[Perimeter](#def-g4-measure-perimeter): $4 \times 85 = 340$ m; minus the gate: $340 - 4 = 336$ m of fence.

**Exercise 28.11 ★★.**

True or false, with grid examples: “if two figures have the same [perimeter](#def-g4-measure-perimeter), they have the same [area](#def-g4-measure-area)”; “if two figures have the same [area](#def-g4-measure-area), they have the same [perimeter](#def-g4-measure-perimeter)”. ([Example 28.7](#ex-g4-measure-samearea) and the chapter’s figure will help.)

**Solution of Exercise 28.11.**

Both claims are *false*. Same [perimeter](#def-g4-measure-perimeter), different [areas](#def-g4-measure-area): the $4 \times 2$ rectangle and the L-shape of the chapter’s figure (both [perimeter](#def-g4-measure-perimeter) $12$; [areas](#def-g4-measure-area) $8$ and $6$). Same [area](#def-g4-measure-area), different [perimeters](#def-g4-measure-perimeter): the $4 \times 4$ square and the $8 \times 2$ rectangle (both [area](#def-g4-measure-area) $16$; [perimeters](#def-g4-measure-perimeter) $16$ and $20$).