### Area

Area |
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#### Area is the amount of surface of a shape.

When we refer to a solid such as a cube, the amount of surface is called its **surface area.**

Area is usually measured in the following units:

- Centimetre square (cm²)
- Metre square (m²)
- Kilometer square (km²)

If 1 m = 100 cm then

1m² = 1m x 1 m

therefore 1m² = 100 cm x 100 cm

= 10 000 cm²

The following shows how the units of area are related.

- 10 000 cm² = 1 m²
- 10 000 m² = 1 ha
- 100 m² = 1 are
- 100 are = 1 ha
- 100 ha = 1 km²

We can use the relationships given above to work out the area of specific and combined shapes.

Similarly we can work out the surface area of solids such as cubes, cuboids and cylinders.

## Let's learn how to calculate area

#### Square

#### Rectangle

#### Triangle

**Circle**

**Parallelogram**

#### Trapezium

#### Can this rectangle become a triangle?

Look at line QP.

It splits the **rectangle** into two equal **Triangles**

That’s why the Area of a Triangle is 1/2 b x h

#### A Circle

Area of a circle is given by:

**A = π × r × r**

The line at the center is called **Diameter**.

Diameter cuts the circle into two equal parts.

Each part is called **semi circle**.

A semi circle can be divided into two equal parts to form a **Quadrant**

When diameter is divided into 2, a **Radius** is formed.

Below is a song that teaches you more about circles.

#### Example

Calculate the area of the circles below

#### Solution

**Area of a circle 1 =Pie × r × r**

Pie =`22/7`

= `22/7` × `14/2` cm ×`14/2` cm

=`22/7` × 7cm × 7cm

=22 × 7cm × 1cm

=154cm2

**Area of a circle 2 = Pie × r × r**

Pie =`22/7`

= `22/7 ` × `14/2` cm × `14/2` cm

=`22/7` × 7cm × 7cm

=22 × 7cm × 1cm

=154cm2

The two circle have the same areas.

The first circle has a diameter of 14cm while the second has a radius of 7cm hence a diameter of 14cm

Find the areas of the shaded part of the circle below given that it has a radius of 7cm.

**Solution**

Area of the whole circle

=`22/7` × 7cm × 7cm

=22 × 1cm × 7cm

= 154cm2

However the circle has been divided into 4 equal parts each called a Quadrant

To get areas of the shaded part divide 154cm2 by 4

= `154/4` cm2

=38.5cm2

#### Parallelogram

The above parallelogram is made up of two right angled triangles and a rectangle.

The two opposite sides are equal

b=b

a=a

The area of a parallelogram is given by:

**Base length × height**

Find the area of the shaded regions in the figures below

#### Answer

Area of a parallelogram = base length X height

** a ) ** ** b)**

=9cm × 5cm =10cm × 4cm

= 45 cm squared = 40 cm squared

#### Trapezium

The above trapezium is made up of a rectangle and a right angled triangle.

The area of a Trapezium is given by:

**½ h × sum of the two parallel sides(a+b)**

Study the image below:

What is the area of the shaded region?

**Remember**

The area of a Trapezium is given by:

½ h × sum of the two parallel sides(a+b)

½ × 4m × (6m + 8m)

2m × (14m)

=28 metres squared.

1. The area of the figure below is 360m2.

Find the length of a.

2. The figure below represents a plot of land. What is the area of the plot.

#### Answers

1. 60 m

2. 9000 metres squared

The figure below shows a circle of radius 7cm touching the vertices of a square.

What is the area of the shaded part?

**Get the area of the circle A = π r^2**

= 3.14 × 7 cm × 7 cm

= 153.86 cm squared

**Get the area of the square.** The radius gives half the length of the side of the square.

To get full length= 7cm × 2

=14cm

Area of a square =S × S

=14cm × 14cm

=196cm squared

**Area of the shaded part = Area of the circle - Area of the square**

= 196cm squared- 153.86cm squared

= 42.14cm squared

Find the area of the shaded part in the figure below

Take pie = 3.14

The shaded part is made up of a semicircle and a right angled triangle

**Area of the semicircle = (πr^2)/2**

80cm = diameter

radius =80/2

= 40cm

3.14 x 40cm x 40cm

=5024 cm2/2

=2512cm squared

**Area of the triangle = ½ b x h**

= ½ x 100cm x 80cm

=50cm x 80cm

= 4000cm squared

**Area of the shaded part = **2512cm squared+ 4000cm squared

= 6512cm squared

The image shows a window of a building in Nairobi.

Calculate the area of the window.

#### Solution

Area = area of the rectangle + area of the semicircle

Area of rectangle= l × w

14 m×7 m

=98m2

Area of the semicircle = (πr^2)/2

22/7 ×7 m×7 m

= 154 m2/2

=77 m2

Answer = 98 m2 + 77 m2

=175 m2

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