13.1 Area of a polygon (EMA7K)

Area

Area is the two dimensional space inside the boundary of a flat object. It is measured in square units.

The acre and the hectare are two common measurements used for the area of land. One hectare is about 0,01 square kilometres and one acre is about 0,004 square kilometres.

Name

Shape

Formula

Square

c0a1ce8d32c6008af7c18b178619cedf.png

area (A)=s2

Rectangle

7f3718335ad257fc182ba01c215a679a.png

area (A)=b×h

Triangle

09e2db9561fc350100eab23c68ccf184.png

area (A)=12b×h

Trapezium

10bdc99a60dea3976d0a7c7983fd449d.png

area (A)=12(a+b)×h

Parallelogram

c3ab06a5756adcffd613de1707aa6cbb.png

area (A)=b×h

Circle

d47b8fd833e895b3a32e01f7fdb0f876.pngarea (A)=πr2 (circumference=2πr)

The video below shows some examples of calculations involving the area of a circle.

Video: 2GRN

Worked example 1: Finding the area of a polygon

Find the area of the following parallelogram:

3b65bb9d743b007da3d605294a6b28d3.png

Find the height BE

AB2=BE2+AE2 (Pythagoras)BE2=AB2AE2=5232=16BE=4 mm

Find the area using the formula for a parallelogram

area=b×h=AD×BE=7×4=28 mm2

The following Phet simulation allows you to build different shapes and calculate the area and perimeter for the shapes: Phet: area builder.

Exercise 13.1

Find the area of each of the polygons below:

4aa1fcf3763efd1633ed8eaeba844d93.png

A=12b×h=12(10)(5)=25 cm2

e8526a33009ba9dcb282713aa66f2f57.png

A=b×h=(10)(5)=50 cm2

2dd90ea62f5b4fc64d8caeb2794fcc2d.png

The radius is half the diameter, therefore the radius is 5 cm.

A=πr2=π(5)2=78,5398...78,54 cm2

527a987a606d7c35033b3baa7e6b9391.png

We first need to work out the height using the theorem of Pythagoras:

h2=5232=16h=4 cm

Now we can calculate the area:

A=b×h=(10)(4)=40 cm2
3452906ba8d4335317cdba933e4d9ef6.png

We first need to work out the height using the theorem of Pythagoras:

h2=10282=36h=6 cm

Now we can calculate the area:

A=12b×h=12(6)(20)=60 cm2
3ef9150b0cda75fbbb58d1ee4b5ece9c.png

We first need to construct the vertical (or perpendicular) height. For an isosceles triangle if we construct the perpendicular height between the two equal sides then this line will bisect the third side.

1f8d3abca4b5a32cd6856a6b1cee077d.png

Now we can calculate the height using the theorem of Pythagoras:

h2=5232=16h=4 cm

Now we can calculate the area:

A=12b×h=12(6)(4)=12 cm2
d41132e03bec282db67a54961399be22.png

We first construct the vertical (perpendicular) height. For an equilateral triangle the perpendicular height will bisect the third side.

bbe98936b7207c7c991d02b406883279.png

Now we can calculate the height using the theorem of Pythagoras:

h2=10252=75h=75 cm

Now we can calculate the area:

A=12base×heightA=12(10)(75)A=43,30 cm2
79feb199e8b5d00088cfef3f5db41cd7.png

We first find the height using the theorem of Pythagoras:

h2=15292=144h=12

Now we can calculate the area:

A=12(a+b)h=12(16+(21+9))(12)=12(46)(12)A=276 cm2

Find an expression for the area of this figure in terms of z and π. The circle has a radius of 3z2. Write your answer in expanded form (not factorised).

e9d46c88f3174abbb23eae182ba4f0ac.png
A=πr2=π(3z2)2=9πz2+12πz+4π

Find an expression for the area of this figure in terms of z and h. The height of the figure is h, and two sides are labelled as 3z2 and z. Write your answer in expanded form (not factorised).

b63544cae4853c9262bfc1e8d6180bb9.png
A=h2(a+b)=h2((3z2)+(z))=2hzh

Find an expression for the area of this figure in terms of x and π. The circle has a radius of x+4. Write your answer in expanded form (not factorised).

69b1042ad35cce69efc8d98b6e86b113.png
A=πr2=π(x+4)2=πx2+8πx+16π

Find an expression for the area of this figure in terms of x and h. The height of the figure is h, and two sides are labelled as x+4 and 3x. Write your answer in expanded form (not factorised).

9d5453502214b09f671dcf1306f2e3b5.png
A=h2(a+b)=h2((x+4)+(3x))=hx+2h