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?

trianglerectangle 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

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

Triangles in real world Do you know…? The Bermuda Triangle is also well known as Devil Triangle or Lost Limbo. It is an area of 3, 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: miles – Distance from Florida to Bermuda: 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 Circumferences Look 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 Perimeter

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 sign A Christmas wreathAn eclipse

How would you calculate the area of a circular crown?

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 chart The 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 that the 25% of the surveyed people is vegetarian.

How to read a pie chart If 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

An 11.11% of the surveyed people prefer pork meat.

How to read a pie chart Put 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

Put in practice: Calculate the angles of these circular sectors.

To conclude… Revise some of your geometric vocabulary: Equilateral triangle Isosceles triangle Scalene triangle Trapezoid Radius Diameter Chord Arc Secant Tangent Parallel lines Angle Right angle Concentric circumferences Diagonal Perimeter Right-angled triangle Acute angle Obtuse angle Hypotenuse Cathetus Area Circular sector Circular crown Volume Apothem Height Base Side

To be continued…