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Area of a polygon
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True or false? All regular polygons can be divided into as many triangles as the polygon has sides.
Maria is going to lay a new floor in her kitchen. She has decided on a pattern of regular hexagons, and has calculated that the floor of the kitchen is 24 square metres. How many floor tiles does she need to buy? First, Maria needs to know the area of each tile. The tiles are regular hexagons, in other words, regular polygons.
A polygon is a two-dimensional geometric figure that has three or more straight sides. Each tile, each hexagonal polygon, can be divided into triangles. Like this: Draw a straight line from each corner into the centre of the polygon. One,two, three, four, five .. six straight lines.
The lines divide the polygon into six triangles, which are the same size, and have the same shape. So it is enough to calculate the area of one of them, and multiply by six. Maria measures the height of one of the triangles .. ..12 centimetres. Then she measures the base of the same triangle ..10 centimetres. The area of the triangle is the base times the height, divided by two.
In other words, 12 centimetres by 10 centimetres divided by two, 60 square centimetres. The total area of the six triangles is then: 60 times six, equals 360 square centimetres. Good! Then we have the area of one of the floor tiles. The tiles should be placed next to each other with no space between.
To know how many tiles are needed, Maria has to compare the area of one tile with the area of the floor. To do so, they must both be expressed in the same units. The floor is 24 square metres. One square metre is 100 by 100 centimetres. That is 10 000 square centimeters.
24 times 10 000 square centimetres is 240 000 square centimetres. NOW the units are the same. Both the tiles and the floor are expressed in square centimetres. 240 000 square centimetres is to be covered by tiles that each have an area of 360 square centimetres. Maria calculates the number of tiles by dividing the area of the kitchen by the area of one tile.
240 000 divided by 360. If we round up to the nearest decimal.. there are 670 floor tiles. So: All regular polygons can be divided into as many triangles as the polygon has sides. Then the area of the polygon can be determined by calculating the area of one of the triangles and multiplying this by the number of sides that the polygon has.
Maria needs at least 670 tiles for her new kitchen floor.