Polygons and circular plane shapes in real life

The Geometry of your life

(part one)

Polygons

The simplest polygon is the triangle. Other polygons can be broken into two or more triangles. Do you remember their names?

triangle rectangle square

pentagon hexagon

Which polygons can you identify in these pictures?

Do you know the name of these shapes?

Areas and Perimeters

•The perimeter of a shape is the total length of its boundaries.

• The area of a shape is the size of the surface which is enclosed by its boundaries .

Look at the example:

This is the boundary…

…and this is the surface.

Area formulae revision

2heightbase sidesideheightbase2

apothemperimeter height

2baseBase

Food for thought…How would you calculate the area of these shapes? Remember that the polygons can be broken into triangles.

Triangles in real worldDo you know…?

The Bermuda Triangle is also well known as Devil Triangle or Lost Limbo. It is an area of 3,900.000 square kilometers, located between Bermuda Island, Puerto Rico and Melbourne (Florida). Several aircrafts and surface vessels disappeared under mysterious circumstances.

Imagine a cruiser which wants to go around the Bermuda Triangle. What is the total distance that it has to go? Express the solution in kilometers.

This is the real surface Work with this approximation

• The perimeter is the sum of the sides’ measures.

• You need some information to calculate it:– Distance from Florida to Puerto Rico:

1,000 miles– Distance from Bermuda to Puerto Rico:

949.38 miles– Distance from Florida to Bermuda:

953.13 miles – Remember : 1 mile = 1.6 kilometers

The red line shows the route from Bermuda Island to the Turks and Caicos Islands, and it represents the height of the triangle. Do you know how to calculate this distance?

Circular Plane Shapes

The circumference and the circle

The circumference is the closed curved line of the points whose distance to the centre (radius) is the same.

The circle is the set of points of the plane enclosed by the circumference boundary.

Concentric CircumferencesLook at the circumferences on the cake surface. What do they have in common?

Two or more circumferences are concentric if they have got the same centre.

Special Effects

• Draw two identical circumferences, as if they were bicycle wheels.

• Draw concentric circumferences to each of the former ones.

• Now, if you move the sheet in a circle, you can notice how the circumferences seem to be two wheels in motion.

Perimeter and area of a circumference

Area

rπ2P Perimeter

2rπA

Can you describe this picture?

Circular Crowns

The surface enclosed by two concentric circumferences is called a circular crown. Look at these examples…

A traffic signA Christmas wreathAn eclipse

How would you calculate the area of a circular crown?

22 rR ππ

2Rπ2rπ

Circular Sectors

A circular sector is the surface of a circumference enclosed by two radii.

Circular sectors in real life

You can see circular sectors in statistics pie charts.

How to read a pie chartThe angle of every circular sector is directly proportional to the percentage it represents. Look at the pie chart.

The angle of the vegetarian sector

is 90º, just a quarter of the pie, what means thatthe 25% of the

surveyed people is vegetarian.

How to read a pie chartIf all the surveyed

people were vegetarian, all the pie chart

would be displayed in blue

color and the angle of the

vegetarian sector would be 360º.

The 100% corresponds to an

angle of 360º.

vegetarian

Then, to calculate the percentage represented by a sector of given angle a ,you only have to do a rule of three like this:

How to read a pie chart

x%aº

100%360º

360100a

x

How to read a pie chart

x%40º

100%360º

11.11%360

10040x

An 11.11% of the surveyed people prefer pork meat.

How to read a pie chartPut in practice: what’s the percentage of people who prefer beef?

To calculate the angle of a sector, given the percentage p, you only have to do a rule of three like this:

How to make a pie chart

xºp%

360º100%

100360p

x

How to make a pie chartPut in practice: Calculate the angles of these circular sectors.

To conclude…

Revise some of your geometric vocabulary:

Equilateral triangle

Isosceles triangle

Scalene triangle

TrapezoidRadius

Diameter

Chord

Arc Secant

Tangent

Parallel lines Angle

Right angle

Concentric circumferences

Diagonal

Perimeter

Right-angled triangle

Acute angle

Obtuse angle

Hypotenuse

Cathetus

AreaCircular sector

Circular crown

Volume

Apothem

Height

BaseSide

To be continued…