Units of area and converting them; estimating

The plane and space, grade 10 (3 units) Lesson 10 of 15Learning

Units of area and converting them; estimating

A square metre in square centimetres

A metre is 100 cm, but a square metre is not 100 cm².

A square of 1 m by 1 m is a square of 100 cm by 100 cm.
So it holds 100 rows, each of 100 squares of 1 cm²: $\displaystyle {100\cdot100}={10{,}000}$ cm².

\displaystyle 1\text{ m}^2=100\cdot100\text{ cm}^2=10000\text{ cm}^2

Every conversion of a unit of length comes into an area twice: once in the length and once in the width.

The dunam and the km²

Large areas – plots, fields, towns – are measured in large units.

Unit	In m²	Example
dunam (the unit of land in Israel)	1,000 m²	a plot for a house with a garden
km² – a square of 1 km by 1 km	$\displaystyle {1{,}000\cdot1{,}000}={1{,}000{,}000}$ m², that is 1,000 dunams	a neighbourhood

Example. A plot of 2.5 dunams: $\displaystyle {2.5\cdot1{,}000}={2{,}500}$ m².
Example. A field of 400 m by 150 m: $\displaystyle {400\cdot150}={60{,}000}$ m², and $\displaystyle {60{,}000\div1{,}000}={60}$ dunams.

Converting through the lengths

It is usually easier to convert the lengths first, and only then multiply.

Example. A tile of 30 cm by 30 cm – what is its area in m²?

The lengths in metres: 0.3 m and 0.3 m.
The area: $\displaystyle {0.3\cdot0.3}={0.09}$ m².
A check: $\displaystyle {30\cdot30}={900}$ cm², and $\displaystyle {900\div10{,}000}={0.09}$ m².

Exercise 1

Convert.

2.4 m² in cm²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

45,000 cm² in m²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

3 dunams in m²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

0.6 km² in dunams

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A tile is a rectangle of 20 cm by 25 cm.

The area of the tile, in m²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How many such tiles cover one m²?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Estimating

An estimate is a quick calculation with round numbers. It gives an answer roughly, and helps to catch a mistake in the exact calculation.

Example. A classroom of 7.8 m by 6.1 m.

Roughly: $\displaystyle {8\cdot6}={48}$ m².
Exactly: $\displaystyle {7.8\cdot6.1}={47.58}$ m² – close to the estimate.
Had the exact calculation given 475.8, the estimate would show at once that the decimal point is in the wrong place.

Exercise 3

A sports hall is a rectangle of 19.6 m by 10.2 m.

Estimate the area of the hall, with round numbers.

The exact area, in m²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A field is a rectangle of 250 m by 80 m.
What is its area, in dunams?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 10 (3 units)

Open as a page

Perimeters

The perimeter of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2(a+b)$
The perimeter of a square – $\displaystyle a$ the side
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4a$
Pi
$\displaystyle \pi\approx3.14$
The circumference – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$

Routes: Pythagoras, distance, speed and time

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2$
The hypotenuse
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2}$
A leg
$\displaystyle a$ $\displaystyle =$ $\displaystyle \sqrt{c^2-b^2}$
Speed – $\displaystyle s$ the distance, $\displaystyle t$ the time
$\displaystyle v$ $\displaystyle =$ $\displaystyle \frac{s}{t}$
Distance
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$
Time
$\displaystyle t$ $\displaystyle =$ $\displaystyle \frac{s}{v}$

Areas

The area of a rectangle
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$
A square metre in square centimetres (a dunam is 1000 m²)
$\displaystyle 1\text{ m}^2=100\cdot100\text{ cm}^2=10000\text{ cm}^2$

Tilings

The sum of the angles of a polygon with $\displaystyle n$ sides
$\displaystyle 180^{\circ}\cdot(n-2)$
The angle of a regular polygon with $\displaystyle n$ sides
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \frac{180^{\circ}\cdot(n-2)}{n}$
The angles around a vertex of a tiling
$\displaystyle \alpha_1+\alpha_2+\ldots=360^{\circ}$