luas yr 4

Area Of Shapes.

8cm

2cm

5cm

3cmA1

A2

16m

12m

10m

12cm

7cm A1 A2

What Is Area ?

Area is the amount of space inside a shape:

Area Area Area Area Area

Area Area Area Area Area

Area Area Area Area Area

Area Area Area Area Area

Area is measured in square centimetres.

1cm

1cm

A square centimetre is a square measuring one centimetre in each direction.

It is written as : 1cm2

1cm2

Estimating The Area.Look at the four shapes below and use your judgement to order them from smallest to largest area:

AB

C

D

AB

C

D

To decide the order of areas consider the four shapes again:

To measure the area we must determine how many square centimetres are in each shape:

Each shape is covered by 36 squares measuring a centimetre by a centimetre .We can now see that all the areas are equal at 36cm2 each.

Area Of A Rectangle.Look again at one of the shapes whose area we estimated:

C

What was the length of the rectangle ? 9cm

How many rows of 9 squares can the breadth hold ? 4

We can now see that the area of the rectangle is given by 9 x 4.

The formula for the area of a rectangle is:

Area = Length x Breadth or A = LB for short.

Length

Breadth

We can now calculate the area of each rectangle very quickly:

(1)

A= L x B

A = 6 x 6 =36cm2

(2)

A= L x B

A = 12 x 3 =36cm2

(3)

A= L x B

A = 9 x 4 =36cm2

(4) A= L x B

A = 18 x 2 =36cm2

Example 1

Calculate the area of the rectangle below:

Solution

A = LB

L = 7 B = 4

A = 7 x 4

A = 28cm2

7cm

4cm(1) (2)

3m

5m

This area is in square metres:

1m

1m

Solution

A = LB

L = 3 B = 5

A = 3 x 5

A = 15m2

Example 3.

Calculate the area of the shape above:

8cm

2cm

5cm

3cm

Solution.

Split the shape up into two rectangles:

A1A2

Calculate the area of A1 and A2.

A1

2

5A2 3

6

Area = A1 + A2

Area = ( 2 x 5) + (6 x 3)

Area = 10 + 18

Area = 28cm2

What Goes In The Box ?Find the area of the shapes below :

(1)

8cm

6cm

4.2m

2.7m

(2)

(3)

17cm

8cm

12cm

5cm

48cm2

11.34m2

141cm2

The Area Of A Triangle.Consider the right angled triangle below:

What shape is the triangle half of ?

Rectangle

8 cm

5cm

What is the area of the rectangle?

Area = 8 x 5 = 40 cm2

What is the area of the triangle ?

Area = ½ x 40 = 20cm2

Base

Height

The formula for the area of a triangle is:

Area = ½ x Base x Height

A = ½ BH

Does the formula apply to all triangles ?

Base (B)

Height (H)

Can we make this triangle into a rectangle ?

Yes

The triangle is half the area of this rectangle:

B

HA1

A1

The areas marked A1 are equal.

A2

A2 The areas marked A2 are equal.

For all triangles:

Area = ½ BH

Calculate the areas of the triangles below:

Example 1

10cm

6cm

Solution.

Area = ½ x base x height

base = 10 cm height = 6cm

Area = ½ x 10 x 6

Area = ½ x 60 = 30cm2

Example 2

6.4m

3.2m

Solution.

Area = ½ x base x height

base = 6.4m height = 3.2m

Area = ½ x 6.4 x 3.2

Area = ½ x 20.48 = 10.24m2

Example 3.

16m

12m

10m

Calculate the area of the shape below: Solution.

Divide the shape into parts:

A1A2

Area = A1 + A2

A1A2

12

10

16-12 =4

10

Area = LB + 1/2 BH

Area = 10 x 12 + ½ x 4 x 10

Area = 120 + 20

Area = 140m2

What Goes In The Box ? 2Find the area of the shapes below :

(1)

8cm

10cm

(2)

10.2 m

6.3m

(3)

25m

18m

12m

40cm2

32.13m2

258m2

The Area Of A Trapezium.A Trapezium is any closed shape which has two sides that are parallel and two sides that are not parallel.

We are now going to find a formula for the area of the trapezium:

a

b

h

Divide the shape into parts:

A1A2

A3

Work out the dimensions of the shapes:

A1

b

h A2

a – b

h

Area = A1 + ( A2 + A3 )

Area = b x h + ½ x (a - b) x h

Area = bh + ½ h(a - b)

Area = bh + ½ ah – ½ bh

Area = ½ ah + ½ bh

Area = ½ h ( a + b )

A3

Often common sense is as good as the formula to work out the area of a trapezium.

Example 1

Calculate the area of the trapezium below :

16cm

11cm

13cm

Solution ( Using the formula).

Area = ½ h ( a + b )

a = 16 b =11 h = 13

Area = ½ x 13 x ( 16 + 11 )

Area = ½ x 13 x 27

Area = 175.5cm2

16cm

11cm

13cm

Solution ( Using composite shapes).

Divide the shape into parts:

11

13 13

16 – 11 = 5

Area = rectangle + triangle

Area = LB + ½ BH

Area = (11x 13) + ( ½ x 5 x 13 )

Area = 143 + 32.5

Area = 175.5cm2

Decide for yourself if you prefer the formula or composite shapes.

Example 2

8m14m

10m

Divide the shape into parts:

10

8

10

14 – 8 = 6

Area = rectangle + triangle

Area = LB + ½ B H

A = ( 10 x 8 ) + ( ½ x 6 x 10 )

A = 80 + 30

A = 110 m 2

What Goes In The Box ? 3Find the area of the shapes below :

(1)

20cm

13cm

10cm

2.7m5.4m4.9m

(2)

165cm2

19.85m2 (to 2 d.p)

The Area Of A Circle.Consider the circle below divided into quarters:

We are going to place the quarters as shown to make the shape below

We can fit a rectangle around this shape:

At the moment it is hard to see why this should tell us how to calculate the area of a circle.

Now consider the same circle split into eight parts:

The eight parts are arranged into the same pattern as last time:

This time the shapes fit the rectangle more closely:

L

B

L

B

This time the shapes fit the rectangle more closely:

What length must the breadth B be close to ?

B = r

What length must the length L be close to ?

Half of the circumference of the circle.

If C = 2 r then L = r .

We now have an approximate length and breadth of our rectangle.

r .

r

What is the area of the rectangle ?A = r x r

A = r 2

If the circle was split into more and more smaller segments and the segments arranged in the same pattern, then the parts would become the rectangle shown above.

See “Autograph Extras”, “New”, “Area Of Circle” for further info’.

r Conclusion.The area of a circle of radius r is given by the formula

A = r 2.

Find the area of the circles below:

Example 1.

20 cm

A = r 2

r = 10

A = 3.14 x 10 x 10

A = 314 cm2

Example 2

2.7m

A = r 2

r = 1.35m

A = 3.14 x 1.35 x 1.35

A = 5.72m2 ( to 2 d.p)

A = r 2

2

Example 3

7cm

Find half the area of a circle:

A = 3.14 x 7 x 7

2

A = 76.93cm2

Example 4

12cm

7cm

Split the shape into two areas.

A1 A2

Area = A1 + A2

Area = LB + ½ r 2.

L = 12 B = 7 r = 3.5

A = 12 x 7 + ½ x 3.14 x 3.5 x 3.5

A = 84 + 19.23

A = 103.2cm 2. (to 1 d.p)

What Goes In The Box ? 4Find the area of the shapes below :

(1)

7cm

(2)6.3m

6.7cm

4.2cm

(3)

153.86cm2

31.16m2 ( 2 d.p)

35.1cm 2 ( 1 d.p)