You will certainly remember from Grade 6 that perimeter is the distance roughly the outermost border of something. Area is the size of a level surface that something. In this chapter, you will find out to use various formulae to calculation the perimeter and area that squares, rectangles and also triangles. You will solve difficulties using these formulae, and also you will likewise learn just how to convert in between different systems of area.

You are watching: What is the relationship between area and square units

Perimeter that polygons

The perimeter of a shape is the complete distance roughly the shape, or the lengths the its sides included together. Perimeter (P) is measure up in units such as millimetres (mm), centimetres (cm) and also metres (m).

Measuring perimeters

Use a compass and/or a ruler to measure up the length of every side in figures A to C. Write the dimensions in mm on each figure.

create down the perimeter of each figure.

*
*
*

The complying with shapes consist of arrows that room equal in length.

What is the perimeter that each shape in number of arrows?

If each arrowhead is 30 mm long, what is the perimeter that each form in mm?

*
*
*
*
*
*
*

Perimeter formulae

If the political parties of a square space all \(s\) devices long:

\<\beginalign \textbfPerimeter the square &= s+ s+s+s\\ &= 4 \times s\\ \textor ns &= 4s\endalign\>


*

If the size of a rectangle is \(l\) units and the breadth (width) is \(b\) units:

\<\beginalign \textbfPerimeter of rectangle &= l+l+b+b\\ &=2\times together + 2 \times b\\ \textor P&=2(l+b) \endalign\>


*

A triangle has three sides, so:

\<\beginalign \textbfPerimeter that triangle &= s_1 + s_2 + s_3\\ \textor ns &= s_1 + s_2 + s_3 \endalign\>


*

Applying perimeter formulae

calculation the perimeter the a square if the size of among its political parties is 17,5 cm.

One next of an equilateral triangle is 32 cm. Calculate the triangle"s perimeter.

Calculate the length of one next of a square if the perimeter the the square is 7,2 m. (Hint: \(4s\ =\) ? because of this \(s =\) ?)

Two sides of a triangle are 2,5 centimeter each. Calculation the length of the 3rd side if the triangle"s perimeter is 6,4 cm.

A rectangle is 40 cm long and also 25 centimeter wide. Calculate its perimeter.

calculation the perimeter of a rectangle the is 2,4 m vast and 4 m long.

The perimeter the a rectangle is 8,88 m. Just how long is the rectangle if it is 1,2 m wide?

perform the essential calculations in your exercise book in order to complete the table. (All the measurements refer to rectangles.)

(b)

(f)

Length

Breadth

Perimeter

(a)

74 mm

30 mm

25 mm

90 mm

(c)

1,125 cm

6,25 cm

(d)

5,5 cm

22 cm

(e)

7,5 m

3,8 m

2,5 m

12 m

Area and also square units

The area that a shape is the size of the flat surface surrounded by the border (perimeter) of the shape.

Usually, area (A) is measure up in square units, such as square millimetres (mm2), square centimetres (cm2) and square metres (m2).

Square units to measure up area

Write under the area of numbers A to E listed below by counting the square units. (Remember to add halves or smaller components of squares.)


*

each square in the grid listed below measures 1 cm2 (1 centimeter \(\times\) 1 cm).

What is the area that the shape drawn on the grid?

top top the very same grid, attract two forms of your own. The shapes should have the exact same area, yet different perimeters.

*

Conversion that units

The figure listed below shows a square through sides of 1 cm.The area that the square is one square centimetre (1 cm2).

How numerous squares of 1 mm by 1 mm (1 mm2) would fit right into the 1 cm2 square? ______ Complete: 1 cm2 = _______ mm2


To readjust cm2 come mm:2

1 cm=2 1 centimeter \(\times\) 1 cm

= 10 mm \(\times\) 10 mm

= 100 mm2

Similarly, to adjust mm2 to cm2:

1 mm2 = 1 mm \(\times\) 1 mm

= 0,1 cm \(\times\) 0,1 cm

= 0,01 cm2

We deserve to use the same method to convert in between other square devices too. Complete:

From m2 come cm2:

\< \beginalign 1 \text m^2 &= 1 \text m \times 1 \text m \\ &=\text______ cm \times \text______ cm\\ &=\text______ cm^2 \endalign\>

From cm2 to m2:

\< \beginalign 1 \text cm^2 &= 1 \text cm \times 1 \text cm \\ &=0.01 \text m \times 0.01\text m\\ &=\text______ m^2 \endalign\>


So, come convert between m2, cm2 and also mm2 you do the following:

cm2 come mm2 \(\rightarrow\) main point by 100 m2 come cm2 \(\rightarrow\) main point by 1000 mm2 to cm2 \(\rightarrow\) divide by 100 cm2 to m2 \(\rightarrow\) divide by 10000

Do the important calculations in your practice book. Then fill in your answers.

15 m2 = ______ cm2 5 cm2 = ______ mm2 20 cm2 = ______ m2 20 mm2 = ______ cm2 25 m2 = ______ cm2 240 000 cm2 = ______ m2 460,5 mm2 = _______ cm2 0,4 m2 = ______ cm2 12 100 cm2 = ______ m2 2,295 cm2 = ______ mm2

Area of squares and rectangles

Investigating the area the squares and rectangles

Each of the adhering to four figures is separated into squares of same size, specific 1 centimeter by 1 cm.


Give the area that each number in square centimetres (cm2):

Area the A:

Area of B:

Area the C:

Area that D:

Is there a shorter an approach to work out the area of every figure? Explain.

number BCDE is a rectangle and MNRS is a square.

*

How numerous cm2 (1 cm \(\times\) 1 cm) would fit right into rectangle BCDE?

How countless mm2 (1 mm \(\times\) 1 mm) would certainly fit right into rectangle BCDE?

What is the area the square MNRS in cm2?

What is the area the square MNRS in mm2?

Figure KLMN is a square with sides of 1 m.


How numerous squares v sides that 1 cm would certainly fit follow me the size of the square?

How many squares v sides of 1 cm would fit along the breadth of the square?

How countless squares (cm2) would therefore fit right into the totality square?

Complete: 1 m2 = ______ cm2


A quick method of calculating the variety of squares that would fit right into a rectangle is to main point the number of squares that would fit follow me its size by the variety of squares that would certainly fit follow me its breadth.


Formulae: area the rectangles and also squares

In the rectangle top top the below: \< \beginalign \textNumber of squares &= \textSquares follow me the length \times \textSquares follow me the breadth \\ &= 6 \times 4 \\ &= 24 \endalign\>

*

From this we deserve to deduce the following:


\< \beginalign \textbfArea of rectangle &= \textLength that rectangle \times \textBreadth the rectangle\\ A &= l \times b\endalign\> where \(A\) is the area in square units, \(l\) is the length and \(b\) is the breadth)

\< \beginalign \textbfArea that square &= \textLength the side \times \textLength that side\\ A &= l \times l \\ &=l^2 \endalign \> wherein \(A\) is the area in square units, and \(l\) is the size of a side)


The devices of the values used in the calculations have to be the same. Remember:

1 m = 100 cm and also 1 cm = 10 mm 1 cm2 = 1 centimeter \(\times\) 1 centimeter = 10 mm \(\times\) 10 mm = 100 mm2 1 m2 = 1 m \(\times\) 1 m = 100 centimeter \(\times\) 100 centimeter = 10 000 cm2 1 mm2 = 1 mm \(\times\) 1 mm = 0,1 centimeter \(\times\) 0,1 cm = 0,01 cm2 1 cm2 = 1 cm \(\times\) 1 cm = 0,01 m \(\times\) 0,01 m = 0,0001 m2 examples

calculate the area that a rectangle through a length of 50 mm and a breadth of 3 cm. Provide the answer in cm2.

Solution:

\< \beginalign \textArea the rectangle & = together \times b & & &\\ &= (50 \times 30) \text mm^2& \text or A &= (5 \times 3)\text cm^2\\ &= 1 500 \text mm^2 & \text or & = 15 \text cm^2 \endalign \>

calculate the area the a square toilet tile with a next of 150 mm.

Solution: \< \beginalign \textArea of square tile &= l \times l \\ &=(150 \times 150) \text mm^2\\ &= 22500\text mm^2\\ \endalign\>

The area is thus 22 500 mm2 (or 225 cm2).

calculation the size of a rectangle if its area is 450 cm2 and also its width is 150 mm.

Solution: \< \beginalign \textArea of rectangle & = together \times b & & &\\ 450 &= l \times 15 & & &\\ 30 \times 15 &= l \times 15 & \text or 450 \div 15& = l\\ 30 = together & & 30 &= l\\ \endalign \>

The size is therefore 30 centimeter (or 300 mm).

Applying the formulae

Calculate the area of every of the complying with shapes:

a rectangle through sides of 12 cm and also 9 cm

a square through sides that 110 mm (answer in cm2)

a rectangle v sides the 2,5 cm and also 105 mm (answer in mm2)

a rectangle through a length of 8 cm and a perimeter the 24 cm

A rugby field has a length of 100 m (goal article to score post) and a breadth the 69 m.

What is the area of the ar (excluding the area behind the score posts)?

What would certainly it cost to plant brand-new grass on the area at a cost of R45/m2?

Another unit for area is the hectare (ha). The is mainly used for measuring land. The dimension of 1 ha is the equivalent of \( 100 \textm \times 100 \textm\). Is a rugby field greater or smaller sized than 1 ha? explain your answer.

execute the crucial calculations in your exercise publication in bespeak to complete the table. (All the dimensions refer to rectangles.)

Length

Breadth

Area

(a)

m

8 m

120 m2

(b)

120 mm

mm

60 cm2

(c)

3,5 m

4,3 m

m2

(d)

2,3 cm

cm

2,76 cm2

(e)

5,2 m

460 cm

m2

figure A is a square v sides that 20 mm. The is reduced as shown in A and also the components are merged to kind figure B. Calculation the area of figure B.


What is the area that the vegetables patch?

She tree carrots on half of the patch, and tomatoes and also potatoes top top a 4 minutes 1 of the patch each. Calculation the area spanned by each kind of vegetable?

How lot will she salary to placed fencing about the patch? The fencing expenses R38/m.

mr Allie has to tile a kitchen floor measuring \(5 \textm \times 4 \textm\). The blue tiles he uses each measure up \(40 \textcm \times 20 \textcm\).


How numerous tiles does grandfather Allie need?

The tiles are offered in boxes containing 20 tiles. How many boxes must he buy?

Doubling a side and its result on area

When a next of a square is doubled, will certainly the area that the square additionally be doubled?

The dimension of each square comprising the grid listed below is \(1 \textcm \times 1 \textcm\).

For every square attracted on the grid, label the lengths that its sides.

Write down the area of every square. (Write the answer inside the square.)

Notice the the 2nd square in each pair of squares has a side length that is double the side length of the an initial square.

Compare the areas of the squares in each pair; then complete the following: once the next of a square is doubled, that is area

*

Area of triangles

Heights and bases that a triangle

The height (h) the a triangle is a perpendicular line segment attracted from a vertex come its the contrary side. Opposing side, which forms a appropriate angle v the height, is called the base (b) that the triangle. Any triangle has actually three heights and three bases.

*

In a right-angled triangle, 2 sides are currently at appropriate angles:

*

Sometimes a base must be prolonged outside that the triangle in bespeak to draw the perpendicular height. This is shown in the very first and 3rd triangles below. Keep in mind that the extended part does not type part that the base"s measurement:

*

Draw any height in each of the following triangles. Brand the elevation (h) and also base (b) on every triangle.

Label another collection of heights and also bases on each triangle.

*

Formula: area that a triangle

ABCD is a rectangle with size = 5 cm and breadth = 3 cm. As soon as A and also C are joined, the creates 2 triangles that are equal in area: \(\triangle ABC\) and also \(\triangle ADC\).

*

\(\textArea of rectangle = together \times b\)

\< \beginalign \textArea the \triangle abc \text (or \triangle ADC\text) &= \frac12 \text(Area of rectangle)\\ &= \frac12(l \times b) \endalign \>

In rectangle ABCD, advertisement is that length and also CD is that breadth.

But look in ~ \(\triangle ADC\). Deserve to you check out that ad is a base and CD is that is height?

So rather of saying:

Area the \(\triangle ADC\) or any kind of other triangle \(= \frac12(l \times b)\)

we say:


\< \beginalign \textbfAre of a triangle &= \frac12 \text(base \times \textheight)\\ &=\frac12(b \times h)\\ \endalign \>

In the formula because that the area of a triangle, b way "base" and not "breadth", and also h method perpendicular height.


Applying the area formula

usage the formula to calculate the locations of the following triangles: \(\triangle ABC\), \(\triangle EFG\), \(\triangle JKL\) and also \(\triangle MNP\).

*

*

PQST is a rectangle in each case below. Calculation the area of \(\triangle PQR\) each time.

*
*

R is the midpoint of QS.

See more: How Much Is A Signed Michael Jordan Jersey Worth, Michael Jordan Autographed Jersey For $8,000

*

In \(\triangle ABC\), the area is 42 m2, and also the perpendicular elevation is 16 m. Find the length of the base.