---
title: "Perimeter and Area"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 34
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/34-perimeter-and-area
---

# Chapter 34 — Perimeter and Area

Grade 4 counted grid squares ([Chapter 28](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#ch-g4-measure)); this chapter earns the first [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) *formula* — length times width for the rectangle — and learns the units cm$^2$ and m$^2$. The formulas for triangles and other shapes ripen in [Chapter 43](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#ch-g6-measure) and [Chapter 53](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#ch-g7-areas).

## 34.1 Two different measures

**Example 34.1 (Border vs surface).**

A gardener needs *[perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter)* to buy the fence and *[area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area)* to buy the grass seed. The two do not follow each other: stretching a rectangle thinner and longer can keep its [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) while its [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) grows — compare a $6 \times 2$ and a $4 \times 3$ rectangle (same [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $12$, [perimeters](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) $16$ and $14$).

## 34.2 The rectangle formula

**Proposition 34.2 (Area of a rectangle).**

A rectangle of length $L$ and width $w$ (in the same unit) has [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area)

$$
A = L \times w ,
$$

in square units: cm$^2$ (squares of side $1$ cm), m$^2$ (squares of side $1$ m), …

**Why, by counting.** Cover the rectangle with unit squares: $w$ rows of $L$ squares each, so $L \times w$ squares in total — exactly the multiplication rectangle of [Chapter 16](https://one-course.com/books/math/1/en/chapter/16-multiplication#ch-g3-mult). ∎

![Three rows of six centimeter-squares: the formula L × w is the counting, written once and for all.](https://one-course.com/images/onecourse/chapters/math-1/g5-areas/fig-7d3aa0dfedf3.svg)

*Three rows of six centimeter-squares: the formula $L \times w$ is the counting, written once and for all.*

**Example 34.3.**

A rug measures $2.5$ m by $2$ m: [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $2.5 \times 2 = 5$ m$^2$. A stamp measures $3$ cm by $2.4$ cm: [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $3 \times 2.4 = 7.2$ cm$^2$. Same formula, any unit — as long as both sides use the *same* one.

**Example 34.4 (Square units convert by 100).**

$1$ m $= 100$ cm, but $1$ m$^2 = 100 \times 100 = 10\,000$ cm$^2$: a square meter is a $100$-by-$100$ grid of square centimeters. [Area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) units jump by hundreds, not tens.

## 34.3 Composite figures

**Method 34.5 (Cut, add, subtract).**

For a figure made of rectangles (an L, a T, a frame):

1. cut it into rectangles, or complete it into a big rectangle;
2. compute each rectangular [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) with the formula;
3. add the pieces — or subtract the hole from the big rectangle;
4. check against a rough count of grid squares.

**Example 34.6.**

An L-shaped room: a $6$ m $\times$ $4$ m rectangle with a $2$ m $\times$ $2$ m corner missing.

$$
\text{Area} = 6 \times 4 - 2 \times 2 = 24 - 4 = 20 \text{ m}^2 .
$$

Or cut the L into a $6 \times 2$ strip and a $4 \times 2$ strip: $12 + 8 = 20$ m$^2$ — two roads, one answer.

**Example 34.7 (Half a rectangle).**

Cutting a rectangle along a diagonal gives two triangles of the same [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area): each is *[half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)* the rectangle. A right triangle with legs $6$ cm and $4$ cm therefore has [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $\frac{6 \times 4}{2} = 12$ cm$^2$ — a picture worth remembering for [Chapter 43](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#ch-g6-measure), where it becomes a formula.

![The diagonal cuts the 6 × 4 rectangle into two equal triangles of 12 squares each.](https://one-course.com/images/onecourse/chapters/math-1/g5-areas/fig-17d400b68130.svg)

*The diagonal cuts the $6 \times 4$ rectangle into two equal triangles of $12$ squares each.*

## 34.4 Exercises

**Exercise 34.1 ★.**

Compute the [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) and the [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) of a rectangle $8$ cm $\times$ $5$ cm. Which answer is in cm, which in cm$^2$?

**Solution of Exercise 34.1.**

[Area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area): $8 \times 5 = 40$ cm$^2$. [Perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter): $2 \times (8 + 5) = 26$ cm.

**Exercise 34.2 ★.**

Compute the [areas](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area): a square of side $9$ cm; a rectangle $12$ m $\times$ $7$ m; a rectangle $4.5$ cm $\times$ $6$ cm.

**Solution of Exercise 34.2.**

$9 \times 9 = 81$ cm$^2$; $12 \times 7 = 84$ m$^2$; $4.5 \times 6 = 27$ cm$^2$.

**Exercise 34.3 ★.**

A rectangle has [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $63$ cm$^2$ and length $9$ cm. Find its width, then its [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter).

**Solution of Exercise 34.3.**

Width: $63 \div 9 = 7$ cm. [Perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter): $2 \times (9 + 7) = 32$ cm.

**Exercise 34.4 ★.**

Draw two different rectangles with [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $24$ squares on grid paper, and compute both [perimeters](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter). Same [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) — same [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter)?

**Solution of Exercise 34.4.**

For instance $6 \times 4$ ([perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) $20$) and $8 \times 3$ ([perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) $22$): same [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $24$, different [perimeters](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter).

**Exercise 34.5 ★.**

Convert: $3$ m$^2$ in cm$^2$; $50\,000$ cm$^2$ in m$^2$. (Remember [Example 34.4](#ex-g5-areas-units): by hundreds!)

**Solution of Exercise 34.5.**

$3$ m$^2 = 30\,000$ cm$^2$; $50\,000$ cm$^2 = 5$ m$^2$.

**Exercise 34.6 ★.**

A T-shaped figure is made of a $8 \times 2$ horizontal bar on top of a $2 \times 5$ vertical bar (in cm). Compute its [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) by adding two rectangles.

**Solution of Exercise 34.6.**

Bar: $8 \times 2 = 16$ cm$^2$; stem: $2 \times 5 = 10$ cm$^2$; total $26$ cm$^2$.

**Exercise 34.7 ★.**

A picture frame: a $30$ cm $\times$ $20$ cm rectangle with a $24$ cm $\times$ $14$ cm rectangular window cut out. What [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) of wood does the frame use?

**Solution of Exercise 34.7.**

$30 \times 20 - 24 \times 14 = 600 - 336 = 264$ cm$^2$ of wood.

**Exercise 34.8 ★.**

Compute the [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) of a right triangle with legs $8$ cm and $6$ cm ([half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) a rectangle, [Example 34.7](#ex-g5-areas-halfrect)).

**Solution of Exercise 34.8.**

[Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the $8 \times 6$ rectangle: $48 \div 2 = 24$ cm$^2$.

**Exercise 34.9 ★.**

A rectangular vegetable patch measures $7$ m by $4$ m. Seed costs $2$ per square meter, and fencing $3$ per meter. Compute the cost of the seed, then the cost of the fence.

**Solution of Exercise 34.9.**

Seed: [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $7 \times 4 = 28$ m$^2$, cost $28 \times 2 = 56$. Fence: [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) $2 \times (7 + 4) = 22$ m, cost $22 \times 3 = 66$.

**Exercise 34.10 ★★.**

A corridor floor is $12$ m long and $2$ m wide, and must be covered with square tiles of side $50$ cm. How many tiles are needed? (Convert first, or count tiles along each direction.)

**Solution of Exercise 34.10.**

Two tiles of $50$ cm make a meter: along the $12$ m length, $24$ tiles; along the $2$ m width, $4$ tiles. Total: $24 \times 4 = 96$ tiles.

**Exercise 34.11 ★★.**

Double the sides of a $4$ cm $\times$ $3$ cm rectangle. What happens to its [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter)? To its [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area)? (Compute both before and after — the two answers differ!)

**Solution of Exercise 34.11.**

Before: [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) $2 \times (4 + 3) = 14$ cm, [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $12$ cm$^2$. After (sides $8$ and $6$): [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) $28$ cm, [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) $48$ cm$^2$. The [perimeter](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-perimeter) *doubled*; the [area](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) was multiplied by *four* ($2 \times 2$): lengths and [areas](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#def-g4-measure-area) do not scale the same way.